A method of generating samples from a classical Boltzmann distribution to train a Restricted Boltzmann Machine may be summarized as including: collecting one or more samples from a physical thermal annealer, wherein the physical thermal annealer is communicatively coupled to a digital computer; receiving by the digital computer the one or more samples from the physical thermal annealer; applying by the digital computer classical Monte Carlo post-processing to the one or more samples to generate one or more post-processed samples. In some embodiments, the method includes applying by the digital computer annealed importance sampling to the one or more post-processed samples.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

In the drawings, identical reference numbers identify similar elements or acts. The sizes and relative positions of elements in the drawings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not necessarily drawn to scale, and some of these elements are arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn are not necessarily intended to convey any information regarding the actual shape of the particular elements, and have been selected for ease of recognition in the drawings.

FIG. 1 is a schematic diagram that illustrates an exemplary hybrid computer including a digital computer and an analog computer in accordance with the present systems, devices, articles, and methods.

FIG. 2 is a schematic diagram that illustrates a portion of an exemplary superconducting quantum processor, suitable for implementing the analog computer of FIG. 1, designed for quantum annealing in accordance with the present systems, devices, articles, and methods.

FIG. 3 is a flow-diagram that illustrates a method for post-processing samples from a physical quantum annealer, in accordance with the present systems, devices, articles, and methods.

FIG. 4A is a graph showing an evolution of an analog processor where the normalized evolution coefficient increases over time.

FIG. 4B is a graph showing an exemplary evolution of an analog processor where the normalized evolution coefficient increases and decreases over time during the course of an annealing schedule.

FIG. 5 is a graph of an exemplary evolution of an analog processor where the analog processor evolves backwards and forwards over time during the course of an annealing schedule.

FIG. 6 is a flow diagram showing a computational method using a hybrid computing system for evolving an analog processor where the analog processor evolves backwards and forwards over time during the course of an annealing schedule.

FIG. 7 is a flow diagram showing a computational method using a hybrid computing system for evolving an analog processor where the analog processor iterates evolving forwards and backwards over time during the course of an annealing schedule.

FIG. 8 is a flow diagram showing a computational method using a hybrid computing system for evolving an analog processor where the analog processor iterates evolving forwards and backwards over time during the course of an annealing schedule without been reprogrammed at each iteration.

FIG. 9 is a graph of an exemplary evolution of an analog processor where the analog processor evolves backwards and forwards over intervals during the course of an annealing schedule.

FIG. 10 is a flow diagram of a computational method using a hybrid computing system for evolving an analog processor where the analog processor evolves forwards and backwards over intervals during the course of an annealing schedule to discover a more suitable annealing schedule for each interval.

FIG. 11 is a graph of an exemplary variation of chain strength over the course of the evolution of a hybrid computing system.

FIG. 12 is a flow diagram of a computational method using a hybrid computing system for evolving an analog processor where the chain strengths of the variables in the analog processor changes over the course of the evolution.

FIG. 13 is a flow diagram of a computational method using a hybrid computing system for evolving an analog processor, where the analog processor evolves backwards and forwards to mitigate the effect of broken chains.

DETAILED DESCRIPTION

In the following description, some specific details are included to provide a thorough understanding of various disclosed embodiments. One skilled in the relevant art, however, will recognize that embodiments may be practiced without one or more of these specific details, or with other methods, components, materials, etc. In other instances, well-known structures associated with quantum processors, such as quantum devices, couplers, and control systems including microprocessors and drive circuitry have not been shown or described in detail to avoid unnecessarily obscuring descriptions of the embodiments of the present methods. Throughout this specification and the appended claims, the words “element” and “elements” are used to encompass, but are not limited to, all such structures, systems, and devices associated with quantum processors, as well as their related programmable parameters.

Unless the context requires otherwise, throughout the specification and claims which follow, the word “comprise” and variations thereof, such as, “comprises” and “comprising” are to be construed in an open, inclusive sense, that is as “including, but not limited to.”

Reference throughout this specification to “one embodiment” “an embodiment”, “another embodiment”, “one example”, “an example”, “another example”, “one implementation”, “another implementation”, or the like means that a particular referent feature, structure, or characteristic described in connection with the embodiment, example, or implementation is included in at least one embodiment, example, or implementation. Thus, the appearances of the phrases “in one embodiment”, “in an embodiment”, “another embodiment” or the like in various places throughout this specification are not necessarily all referring to the same embodiment, example, or implementation. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments, examples, or implementations.

It should be noted that, as used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the content clearly dictates otherwise. Thus, for example, reference to a problem-solving system including “a quantum processor” includes a single quantum processor, or two or more quantum processors. It should also be noted that the term “or” is generally employed in its sense including “and/or” unless the content clearly dictates otherwise.

The headings provided herein are for convenience only and do not interpret the scope or meaning of the embodiments.

Hybrid Computing System Comprising a Quantum Processor

FIG. 1 illustrates a hybrid computing system 100 including a digital computer 102 coupled to an analog computer 104. In some implementations, the analog computer 104 is a quantum computer and the digital computer 102 is a classical computer.

The exemplary digital computer 102 includes a digital processor (such as one or more central processor units 106) that may be used to perform classical digital processing tasks described in the present systems and methods. Those skilled in the relevant art will appreciate that the present systems and methods can be practiced with other digital computer configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, personal computers (“PCs”), network PCs, mini-computers, mainframe computers, and the like, when properly configured or programmed to form special purpose machines, and/or when communicatively coupled to control an analog computer, for instance a quantum computer.

Digital computer 102 will at times be referred to in the singular herein, but this is not intended to limit the application to a single digital computer. The present systems and methods can also be practiced in distributed computing environments, where tasks or sets of instructions are performed or executed by remote processing devices, which are linked through a communications network. In a distributed computing environment computer- or processor-readable instructions (sometimes known as program modules), application programs and/or data, may be located in both local and remote memory storage devices (e.g., nontransitory computer- or processor-readable media).

Digital computer 102 may include at least one or more digital processors (e.g., one or more central processor units 106), one or more system memories 108, and one or more system buses 110 that couples various system components, including system memory 108 to central processor unit 106.

The digital processor may be any logic processing unit, such as one or more central processing units (“CPUs”) with one or more cores, graphics processing units (“GPUs”), digital signal processors (“DSPs”), application-specific integrated circuits (“ASICs”), field-programmable gate arrays (“FPGAs”), programmable logic controllers (PLCs), etc.

Digital computer 102 may include a user input/output subsystem 112. In some implementations, the user input/output subsystem includes one or more user input/output components such as a display 114, mouse 116, and/or keyboard 118. System bus 110 can employ any known bus structures or architectures, including a memory bus with a memory controller, a peripheral bus, and a local bus. System memory 108 may include non-volatile memory, for example one or more of read-only memory (“ROM”), static random access memory (“SRAM”), Flash NAND; and volatile memory, for example random access memory (“RAM”) (not shown), all of which are examples of nontransitory computer- or processor-readable media.

A basic input/output system (“BIOS”) 120, which can form part of the ROM, contains basic routines that help transfer information between elements within digital computer 102, such as during startup.

Digital computer 102 may also include other non-volatile memory 122. Non-volatile memory 122 may take a variety of forms, including: a hard disk drive for reading from and writing to a hard disk, an optical disk drive for reading from and writing to removable optical disks, and/or a magnetic disk drive for reading from and writing to magnetic disks, all of which are examples of nontransitory computer- or processor-readable media. The optical disk can be a CD-ROM or DVD, while the magnetic disk can be a magnetic floppy disk or diskette. Non-volatile memory 122 may communicate with digital processor via system bus 110 and may include appropriate interfaces or controllers 124 coupled to system bus 110. Non-volatile memory 122 may serve as nontransitory long-term storage for computer- or processor-readable instructions, data structures, or other data (also called program modules) for digital computer 105.

Although digital computer 102 has been described as employing hard disks, optical disks and/or magnetic disks, those skilled in the relevant art will appreciate that other types of non-volatile computer-readable media may be employed, such a magnetic cassettes, flash memory cards, Flash, ROMs, smart cards, etc., all of which are further examples of nontransitory computer- or processor-readable media. Those skilled in the relevant art will appreciate that some computer architectures conflate volatile memory and non-volatile memory. For example, data in volatile memory can be cached to non-volatile memory, or a solid-state disk that employs integrated circuits to provide non-volatile memory. Some computers place data traditionally stored on disk in memory. As well, some media that are traditionally regarded as volatile can have a non-volatile form, e.g., Non-Volatile Dual In-line Memory Module variation of Dual In Line Memory Modules.

Various sets of computer- or processor-readable instructions (also called program modules), application programs and/or data can be stored in system memory 108. For example, system memory 108 may store an operating system 126, and a set of computer- or processor-readable server instructions (i.e., server modules) 128. In some implementations, server module 128 includes instructions for communicating with remote clients and scheduling use of resources including resources on the digital computer 102 and analog computer 104. For example, a Web server application and/or Web client or browser application for permitting digital computer 102 to exchange data with sources via the Internet, corporate Intranets, or other networks, as well as with other server applications executing on server computers.

In some implementations, system memory 108 may store a set of computer- or processor-readable calculation instructions (i.e., calculation module 130) to perform pre-processing, co-processing, and post-processing to analog computer 104.

In some implementations, system memory 108 may store post-processing instructions, or make use of the instructions in calculation instructions module 130. Execution of the post-processing instructions can cause a processor (such as CPU 106) to perform post-processing in digital computer 102. For example, digital computer 102 can perform post-processing of samples obtained from analog computer 104 based on post-processing instructions in calculation instructions module 130. Post-processing of samples from a physical quantum annealer, such as analog computer 104, is described in following sections of the present disclosure. Post-processing can include, for example, quantum Monte Carlo and/or annealed importance sampling.

In accordance with the present systems and methods, system memory 108 may store at set of analog computer interface modules 132 operable to interact with the analog computer 104.

In some implementations, system memory 108 may store a set of Boltzmann machine instructions or a Boltzmann machine module 134 to provide procedures and parameters for the operation of the analog computer 104 as a Boltzmann machine. For example, the Boltzmann machine module 134 can implement a method (such as method 300 of FIG. 3) on digital computer 102 and analog computer 104. The hybrid computer 100 following instructions in the Boltzmann machine module 134 can implement graphical representations of portions of Boltzmann machines.

In some implementations, system memory includes a set of training and validations instructions or training and validations instructions module 136. A Boltzmann machine can be trained via supervised or unsupervised learning. The hybrid computer 100 may implement training methods defined in the training and validations instructions module 136. As well, a Boltzmann machine once trained may need validating. The hybrid computer 100 may validate a Boltzmann machine following methods defined in the training and validations instructions module 136.

In some implementations, system memory 108 may store a set of runtime instructions or runtime instructions module 138 to provide executable procedures and parameters to deploy and/or monitor a Boltzmann machine.

While shown in FIG. 1 as being stored in system memory 108, the modules shown and other data can also be stored elsewhere including in non-volatile memory 122 or one or more other non-transitory computer- or processor-readable media.

The analog computer 104 can be provided in an isolated environment (not shown). For example, where the analog computer 104 is a quantum computer, the environment shields the internal elements of the quantum computer from heat, magnetic field, and the like. The analog computer 104 includes one or more analog processors 140. Examples of analog processor 140 include quantum processors such as those described below in reference to FIG. 2.

A quantum processor includes programmable elements such as qubits, couplers, and other devices. The qubits are read out via readout system 142. These results are fed to the various sets of computer- or processor-readable instructions for the digital computer 102 including server module 128, calculation module 130, analog computer interface modules 132, or other modules stored in non-volatile memory 122, returned over a network or the like. The qubits are controlled via qubit control system 144. The couplers are controlled via coupler control system 146. In some embodiments, the qubit control system 144 and the coupler control system 146 are used to implement quantum annealing, as described herein, on analog processor 140.

In some implementations, the digital computer 102 can operate in a networked environment using logical connections to at least one client computer system. In some implementations, the digital computer 102 is coupled via logical connections to at least one database system. These logical connections may be formed using any means of digital communication, for example, through a network, such as a local area network (“LAN”) or a wide area network (“WAN”) including, for example, the Internet. The networked environment may include wired or wireless enterprise-wide computer networks, intranets, extranets, and/or the Internet. Other embodiments may include other types of communication networks such as telecommunications networks, cellular networks, paging networks, and other mobile networks. The information sent or received via the logical connections may or may not be encrypted. When used in a LAN networking environment, digital computer 102 may be connected to the LAN through an adapter or network interface card (“NIC”) (communicatively linked to system bus 110). When used in a WAN networked environment, digital computer 102 may include an interface and modem (not shown), or a device such as NIC, for establishing communications over the WAN. Non-networked communications may additionally, or alternatively, be employed.

In accordance with some embodiments of the present systems and devices, a quantum processor (such quantum processor 140) may be designed to perform quantum annealing and/or adiabatic quantum computation. An evolution Hamiltonian is constructed, that is proportional to the sum of a first term proportional to a problem Hamiltonian and a second term proportional to a delocalization Hamiltonian, as follows:

H
E

∝

A
⁡
(
t
)

⁢

H
P

+

B
⁡
(
t
)

⁢

H
D

- where H_Eis the evolution Hamiltonian, H_Pis the problem Hamiltonian, H_Pis the delocalization Hamiltonian, and A(t), B(t) are coefficients that can control the rate of evolution, and typically lie in the range [0, 1].

In some implementations, a time-varying envelope function is placed on the problem Hamiltonian. A suitable delocalization Hamiltonian is given by:

H
D

∝

-

1
2

⁢

∑

i
=
1

N

Δ
i

⁢

σ
i
x

- where N represents the number of qubits, σ_i^xis the Pauli x-matrix for the i^thqubit and Δ_iis the single qubit tunnel splitting induced in the i^thqubit. Here, the σ_i^xterms are examples of “off-diagonal” terms.

A common problem Hamiltonian includes a first component proportional to diagonal single qubit terms, and a second component proportional to diagonal multi-qubit terms, and may be of the following form:

H
P

∝

-

ε
2

[

∑

i
=
1

N

h
i

⁢

σ
i
z

+

∑

j
>
i

N

J
ij

⁢

σ
i
z

⁢

σ
j
z

]

- where N represents the number of qubits, σ_i^zis the Pauli z-matrix for the i^thqubit, h_iand J_ijare dimensionless local fields for the qubits, and couplings between qubits, respectively, and ε is a characteristic energy scale for H_P.

The σ_i^zand σ_i^zσ_j^zterms are examples of “diagonal” terms. The former is a single qubit term and the latter a two qubit term.

Throughout this specification, the terms “problem Hamiltonian” and “final Hamiltonian” are used interchangeably unless the context dictates otherwise. Certain states of the quantum processor are, energetically preferred, or simply preferred by the problem Hamiltonian. These include the ground states but may include excited states.

Hamiltonians such as H_Dand H_Pin the above two equations, respectively, may be physically realized in a variety of different ways. A particular example is realized by an implementation of superconducting qubits.

Exemplary Superconducting Quantum Processor for Quantum Annealing

FIG. 2 is a schematic diagram of a portion of an exemplary superconducting quantum processor 200 designed for quantum annealing (and/or adiabatic quantum computing) components from which may be used to implement the present systems and devices. The portion of superconducting quantum processor 200 shown in FIG. 2 includes two superconducting qubits 202, and 204. Also shown is a tunable σ_i^zσ_j^zcoupling (diagonal coupling) via coupler 210 therebetween qubits 202 and 204 (i.e., providing 2-local interaction). While the portion of quantum processor 200 shown in FIG. 2 includes only two qubits 202, 204 and one coupler 206, those of skill in the art will appreciate that quantum processor 200 may include any number of qubits and any number of couplers coupling information therebetween.

The portion of quantum processor 200 shown in FIG. 2 may be implemented to physically realize quantum annealing and/or adiabatic quantum computing. Quantum processor 200 includes a plurality of interfaces 208, 210, 212, 214, and 216 that are used to configure and control the state of quantum processor 200. Each of interfaces 208, 210, 212, 214, and 216 may be realized by a respective inductive coupling structure, as illustrated, as part of a programming subsystem and/or an evolution subsystem. Such a programming subsystem and/or evolution subsystem may be separate from quantum processor 200, or it may be included locally (i.e., on-chip with quantum processor 200) as described in, for example, U.S. Pat. Nos. 7,876,248 and 8,035,540.

In the operation of quantum processor 200, interfaces 208 and 214 may each be used to couple a flux signal into a respective compound Josephson junction 218 and 220 of qubits 202 and 204, thereby realizing a tunable tunneling term (the Δi term) in the system Hamiltonian. This coupling provides the off-diagonal σ^xterms of the Hamiltonian and these flux signals are examples of “delocalization signals”.

In some implementations, the tunneling term is selected to make a first portion of the qubits on the quantum processor more classical relative a second portion of the qubits. For example, qubit 202 may be a hidden unit in a Boltzmann machine and have a smaller tunneling term relative to qubit 204.

Similarly, interfaces 210 and 212 may each be used to apply a flux signal into a respective qubit loop of qubits 202 and 204, thereby realizing the hi terms in the system Hamiltonian. This coupling provides the diagonal σ²terms in the system Hamiltonian. Furthermore, interface 216 may be used to couple a flux signal into coupler 206, thereby realizing the J_ijterm(s) in the system Hamiltonian. This coupling provides the diagonal σ_i^zσ_j^zterms in the system Hamiltonian.

In FIG. 2, the contribution of each of interfaces 208, 210, 212, 214, and 216 to the system Hamiltonian is indicated in boxes 208a, 210a, 212a, 214a, and 216a, respectively. As shown, in the example of FIG. 2, the boxes 208a, 210a, 212a, 214a, and 216a are elements of time-varying Hamiltonians for quantum annealing and/or adiabatic quantum computing.

Throughout this specification and the appended claims, the term “quantum processor” is used to generally describe a collection of physical qubits (e.g., qubits 202 and 204) and couplers (e.g., coupler 206). The physical qubits 202 and 204 and the coupler 206 are referred to as the “programmable elements” of the quantum processor 200 and their corresponding parameters (e.g., the qubit h_ivalues and the coupler J_ijvalues) are referred to as the “programmable parameters” of the quantum processor. In the context of a quantum processor, the term “programming subsystem” is used to generally describe the interfaces (e.g., “programming interfaces” 210, 212, and 216) used to apply the programmable parameters (e.g., the h_iand J_ijterms) to the programmable elements of the quantum processor 200 and other associated control circuitry and/or instructions.

As previously described, the programming interfaces of the programming subsystem may communicate with other subsystems which may be separate from the quantum processor or may be included locally on the processor. As described in more detail later, the programming subsystem may be configured to receive programming instructions in a machine language of the quantum processor and execute the programming instructions to program the programmable elements in accordance with the programming instructions. Similarly, in the context of a quantum processor, the term “evolution subsystem” generally includes the interfaces (e.g., “evolution interfaces” 208 and 214) used to evolve the programmable elements of the quantum processor 200 and other associated control circuitry and/or instructions. For example, the evolution subsystem may include annealing signal lines and their corresponding interfaces (208, 214) to the qubits (202, 204).

Quantum processor 200 also includes readout devices 222 and 224, where readout device 222 is associated with qubit 202 and readout device 224 is associated with qubit 204. In some embodiments, such as shown in FIG. 2, each of readout devices 222 and 224 includes a DC-SQUID inductively coupled to the corresponding qubit. In the context of quantum processor 200, the term “readout subsystem” is used to generally describe the readout devices 222, 224 used to read out the final states of the qubits (e.g., qubits 202 and 204) in the quantum processor to produce a bit string. The readout subsystem may also include other elements, such as routing circuitry (e.g., latching elements, a shift register, or a multiplexer circuit) and/or may be arranged in alternative configurations (e.g., an XY-addressable array, an XYZ-addressable array, etc.). Qubit readout may also be performed using alternative circuits, such as that described in PCT Patent Publication WO2012064974.

While FIG. 2 illustrates only two physical qubits 202, 204, one coupler 206, and two readout devices 222, 224, a quantum processor (e.g., processor 200) may employ any number of qubits, couplers, and/or readout devices, including a larger number (e.g., hundreds, thousands or more) of qubits, couplers and/or readout devices. The application of the teachings herein to processors with a different (e.g., larger) number of computational components should be readily apparent to those of ordinary skill in the art.

Examples of superconducting qubits include superconducting flux qubits, superconducting charge qubits, and the like. In a superconducting flux qubit the Josephson energy dominates or is equal to the charging energy. In a charge qubit it is the reverse. Examples of flux qubits that may be used include rf-SQUIDs, which include a superconducting loop interrupted by one Josephson junction, persistent current qubits, which include a superconducting loop interrupted by three Josephson junctions, and the like. See, examples of rf-SQUID qubits in Bocko, et al., 1997, IEEE Trans. on Appl. Supercond. 7, 3638; Friedman, et al., 2000, Nature 406, 43; and Harris, et al., 2010, Phys. Rev. B 81, 134510; or persistent current qubits, Mooij et al., 1999, Science 285, 1036; and Orlando et al., 1999, Phys. Rev. B 60, 15398. In addition, hybrid charge-phase qubits, where the energies are equal, may also be used. Further details of superconducting qubits may be found in Makhlin, et al., 2001, Rev. Mod. Phys. 73, 357; Devoret et al., 2004, arXiv:cond-mat/0411174; Zagoskin and Blais, 2007, Physics in Canada 63, 215; Clarke and Wilhelm, 2008, Nature 453, 1031; Martinis, 2009, Quantum Inf. Process. 8, 81; and Devoret and Schoelkopf, 2013, Science 339, 1169. In some embodiments, the qubits and couplers are controlled by on chip circuitry. Examples of on-chip control circuitry can be found in U.S. Pat. Nos. 7,876,248; 7,843,209; 8,018,244; 8,098,179; 8,169,231; and 8,786,476. Further details and implementations of exemplary quantum processors that may be used in conjunction with the present systems and devices are described in, for example, U.S. Pat. Nos. 7,533,068; 8,008,942; 8,195,596; 8,190,548; and 8,421,053.

Sampling Using a Physical Quantum Annealer

A physical quantum annealer (PQA) can be used to change the Hamiltonian of a quantum system, and can cause a change in a state of the quantum system. After annealing, the Hamiltonian of the quantum system can be similar, or the same, as a problem Hamiltonian H_P. The PQA can be an open system, i.e., a system that interacts with the environment. In the case of an open-system quantum annealer, the state can be at least an approximation to a thermal state of the quantum system. In the special case of an adiabatic quantum annealer (where the system is isolated from the environment), the state can be at least an approximation to the ground state of Hamiltonian H_P. The following paragraphs refer to an open-system quantum annealer.

The state of a PQA at normalized time t (where t∈[0,1]) can be described by a density matrix ρ_ij(t), where i, j denote eigenstates of a Hamiltonian H(t). The state can be modeled by an equation of the form {dot over (ρ)}_ij=−i[H,ρ]+F(ρ), where F(ρ) is a linear matrix-valued function.

At the start of annealing, the density matrix is diagonal and the state of the PQA can be described by a quantum Boltzmann distribution. At an intermediate time during annealing t¹, the state of the quantum system can begin to deviate from the quantum Boltzmann distribution. One reason for the deviation can be a slowdown of open-system quantum dynamics.

The point at which the state begins to deviate from the quantum Boltzmann distribution can be referred to as the freeze-out point. Past that point the state will deviate from quantum Boltzmann distribution. There can be multiple freeze-out points t¹<t²< . . . <tⁿwhere the dynamics between progressively smaller subspaces of the state space slow down. If the points t¹, . . . tⁿare sufficiently close to each other, the state of the PQA in the region t E [t¹,tⁿ] can be close to a quantum Boltzmann distribution.

The time up to which the state of the PQA is close to a quantum Boltzmann distribution can be denoted as t. For normalized time t>t, the state can increasingly deviate from a quantum Boltzmann distribution, and its evolution can be described as “running downhill” in the quantum configuration space, reaching equilibrium locally in subspaces while not necessarily reaching equilibrium globally.

The distribution corresponding to the state of the PQA at normalized time t can be denoted as p(t). For annealing parameters θ(t) at time t, the corresponding quantum Boltzmann distribution can be denoted as p_θ(t)^QB. Samples returned by the PQA correspond to samples from distribution p(1). As described above, the distribution p(t) can be close to p_θ(t)^QB;

It can be impractical to obtain samples from the PQA at time t, so, in practice, samples are typically obtained from the PQA in its final state after annealing.

Sampling from Intermediate Quantum Boltzmann Distributions Using a Physical Quantum Annealer

A physical quantum annealer (PQA), such as a superconducting quantum processor described in reference to FIGS. 1 and 2, can return samples from the final distribution p(1). It can be beneficial to convert the samples from the final distribution p(1) to good-quality samples from an intermediate distribution p(t)≈p_θ(t)^QB; Good-quality samples are samples meeting a determined threshold for closeness to true samples from a distribution. The good-quality samples can be used in applications requiring samples from a quantum Boltzmann distribution.

Furthermore, it can be beneficial to convert the good-quality samples from intermediate distribution p_θ(t)^QBto samples from another quantum Boltzmann distribution p_θ(t)^QB, and/or to samples from a classical Boltzmann distribution.

In previous approaches, the samples from the final distribution p(1) obtained from a PQA were treated as though they came from an unknown distribution, and were post-processed (e.g., using a classical Markov Chain Monte Carlo method) to convert them to a classical Boltzmann distribution. A shortcoming of previous approaches is that little or no use is made of intermediate quantum distribution p_θ(t)^QBwhich contains global information about the final distribution p(1). Previous classical post-processing methods are local, and generally unable to affect global features of the distribution. Consequently, previous approaches to post-processing of samples obtained from a PQA can misrepresent global features of a classical Boltzmann distribution of interest.

Quantum Monte Carlo Post-Processing

The presently disclosed systems and methods can use Quantum Monte Carlo (QMC) post-processing to correct for local bias in samples returned by a PQA. QMC is a method that can be used to obtain samples from a quantum Boltzmann distribution on a classical computer.

QMC post-processing can include taking the final samples x_a, a=1 . . . N from the PQA, and initializing MCMC chains with those samples x_a⁽⁰⁾. Here x denotes a quantum state that is represented, for example, as a path configuration of Path Integral QMC. MCMC chains can be evolved using a QMC transition operator corresponding to the distribution of interest T_θ(t)(x⁽ⁱ⁾,x⁽ⁱ⁺¹⁾). The transition operator can satisfy the following detailed balance condition:

p

θ
⁡
(

t
_

)

QB

(

x

(
i
)

)

⁢

T

θ
⁡
(

t
_

)

(

x

(
i
)

,

x

(

i
+
1

)

)

=

p

θ
⁡
(

t
_

)

QB

(

x

(

i
+
1

)

)

⁢

T

θ
⁡
(

t
_

)

(

x

(

i
+
1

)

,

x

(
i
)

)

.

Running a QMC chain for long enough can yield samples from distribution p_θ(t)^QB. The minimum time needed to obtain such samples starting from random states x_a⁽⁰⁾can be referred to as an equilibration time. Starting the MCMC chains with PQA samples can reduce the equilibration time. One reason can be that the global features of the distribution p_θ(t)^QBare captured more correctly by PQA samples (which can provide relative probabilities of subspaces of the quantum state space).

To convert samples x_ainto equilibrium samples from p_θ(t)^QB, it can be sufficient to equilibrate locally (i.e., within subspaces). Local equilibration can be faster, and can be considered as a post-processing technique. As a result, applying QMC post-processing with M steps can produce good-quality samples x_a^(M)from quantum Boltzmann distribution p_θ(t)^QBfor relatively small M.

In general, freeze-out point t is unknown. One approach is to choose a freeze-out point, and halt the annealing for a determined time at the freeze-out point, before re-starting the annealing. This approach is referred to as “annealing with pause” or “mid-anneal pause”, and is described in International PCT Patent Application Publication No. WO2017075246A1 and U.S. Patent Application Ser. No. 62/331,288 entitled “SYSTEMS AND METHODS FOR DEGENERACY MITIGATION IN A QUANTUM PROCESSOR”.

Another approach to determining t is to compute certain statistics of a quantum Boltzmann distribution p_θ(t)^QBfor various points t∈[0,1], and define t as the point where these statistics are closest to the ones computed from samples obtained from a physical quantum annealer, and post-processed as described above. Such statistics can include spin and spin-spin expectations, average energy, variance of energy, and other suitable statistics. There can be several points where the statistics are close, and these points correspond to multiple freeze-out points. In one implementation, the first of these points is selected as t.

Annealed Importance Sampling to Convert Samples from a Quantum Boltzmann Distribution to another Boltzmann Distribution

The presently disclosed systems and methods include the use of annealed importance sampling to convert samples from good-quality samples of p_θ(t)^QBto samples from another quantum Boltzmann distribution p_θ′^QB.

A sequence of intermediate quantum Boltzmann distributions can be generated as follows:

p

θ
k

QB

,

k
=

1
⁢

…
⁢

L

,

θ
1

=

θ
⁡
(

t
_

)

,

θ
L

=

θ
′

- so that distributions in every pair of consecutive distributions in the above sequence are sufficiently close to one another. Sufficiently close means that importance sampling of a first distribution from a pair of consecutive distributions in the above sequence can be performed efficiently using samples from a second distribution from the pair of consecutive distributions. One approach to selecting parameters θ_kis to linearly interpolate between θ(t) and θ′, and choose L to be large enough that the distributions are sufficiently close.

A sequence of states {right arrow over (x)}=(x¹, x². . . x^L) can be sampled from intermediate distributions

p

θ

t
k

with a probability as follows:

P
⁡
(

x
1

,

x
2

,
…

,

x
L

)

=

T

θ

t
L

(

x
L

,

x

L
-
1

)

⁢

…
⁢

T

θ

t
3

(

x
3

,

x
2

)

⁢

T

θ

t
2

(

x
2

,

x
1

)

⁢

p

θ

t
1

(

x
1

)

.

A weight can be assigned to each sample as follows:

w
⁡
(

x
1

,

x
2

,
…

,

x
L

)

=

p
~

θ

t
2

(

x
1

)

p
~

θ

t
1

(

x
1

)

⁢

p
~

θ

t
3

(

x
2

)

p
~

θ

t
2

(

x
2

)

⁢

…
⁢

p
~

θ

t
L

(

x

L
-
1

)

p
~

θ

t

L
-
1

(

x

L
-
1

)

- where

p
˜

θ

t
k

is an unnormalized probability.

The samples x^Lcan be used to compute an expected value of a function F(x) as follows:

〈

F
⁡
(
x
)

=

∑

x
→

~

P
⁡
(

x
→

)

w
⁡
(

x
→

)

⁢

F
⁡
(

x
L

)

∑

x
→

~

P
⁡
(

x
→

)

w
⁡
(

x
→

)

Efficiency of the approach can be characterized by the number of effective samples, as follows:

N
eff

=

(

∑

x
→

~

P
⁡
(

x
→

)

w
⁡
(

x
→

)

)

2

∑

x
→

~

P
⁡
(

x
→

)

w
⁡
(

x
→

)

2

If distributions p_θ(t)^QBand p_θ′^QBare sufficiently different from one other, the number of effective samples Neff can be small enough that the estimator for F(x) has a relatively high variance. Increasing the number of intermediate distributions L can increase the number of effective samples Neff, and reduce the variance of the estimator for F(x).

Training Quantum Boltzmann Machines and Restricted Boltzmann Machines

FIG. 3 is a flow-diagram that illustrates a method 300 for post-processing samples from a physical quantum annealer, in accordance with the present systems, devices, articles, and methods. One or more of the acts in method 300 may be performed by or via one or more circuits, for instance one or more hardware processors. In some examples, a device including a hybrid computer (such hybrid computer 100 of FIG. 1) performs the acts in method 300.

Method 300 starts at 302, for example in response to an invocation by an invoking program, procedure, routine or function.

At 304, a computational system (e.g., hybrid computer 100 of FIG. 1) collects samples from a physical quantum annealer (PQA). At 306, the computational system applies QMC post-processing to the collected samples.

If, at 308, the computational system determines the post-processed samples are not for input to a Quantum Boltzmann Machine, method 300 proceeds to 310. At 310, the computational system applies annealed importance sampling (AIS) post-processing to the post-processed samples output from the QMC. At 312, method 300 ends.

If, at 308, the computational system determines the post-processed samples are for input to a Quantum Boltzmann Machine, method 300 proceeds to the end at 312.

FIG. 4A is a graph 400a of an evolution of an analog processor over time. An analog processor may be a quantum processor comprising superconducting qubits and couplers.

Vertical axis 402 represents the normalized evolution coefficient s and the horizontal axis 404 represent the time of the evolution of the analog processor. The normalized evolution coefficient s may represent the normalized flux applied to a compound Josephson junction or the normalized persistent current I_Pof a flux qubit. The normalized evolution coefficient s changes monotonically over time, increasing from 0 to a maximum value of 1.

The normalized evolution coefficient can also be referred to as the anneal fraction. The normalized evolution coefficient (or anneal fraction) is a parameter that can vary with time between 0 and 1, and can be used to define an annealing schedule.

A person skilled in the art will understand that the rate of change of the normalized evolution coefficient s over time is shown in FIG. 4A for illustration purposes only and in other implementations the normalized evolution coefficient can increase at a slower or faster rate. In some implementations the normalized evolution coefficient s can change non-linearly. Examples of evolution schedules of analog processors are described in Patent Publication No. US 2015/0363708.

Techniques described herein are used to operate a hybrid processor comprising an analog processor and a digital processor where the normalized evolution coefficient s may increase and/or decrease over the course of the operation of the hybrid processor. For certain operations, it may be desirable to operate the hybrid processor such that the analog processor reaches a predetermined classical spin state at the end of a first or initial evolution. This technique may allow study of problem dynamics, or it may be used for obtaining samples from the analog processor.

FIG. 4B is a graph of an example evolution 400b of an analog processor over time, operating with a digital processor to form a hybrid processor according to the present systems, methods and apparatus. An analog processor may comprise a quantum processor. Vertical axis 402 represents the normalized evolution coefficient s and the horizontal axis 404 the time of the evolution of the analog processor.

Before the start of example evolution 400b, the hybrid processor may determine a classical spin state and apply one or more preparatory biases to the analog processor to target the evolution of the analog processor towards the classical spin state. Preparatory biases may be applied via the analog processor's circuitry components, for example via on-chip DACs or analog lines. Preparatory biases may influence the evolution of the analog processor towards a classical state. When the analog processor is a quantum processor with n qubits, there are 2ⁿclassical states.

In example evolution 400b the normalized evolution coefficient s increases from a value of 0 at time t=0 to a value of 1 at time t₁. A person skilled in the art will understand that the rate of the evolution from time t=0 to t₁is shown in FIG. 4B for illustration purposes only and in other implementations the rate of the evolution of the analog processor from 0 to t₁may be faster or slower than illustrated.

At t₁, the evolution is paused until time t₂. During the time interval between t₁and t₂, shown in FIG. 4B as time interval 406, the digital processor may remove the preparatory biases applied before the start of example evolution 400b. A person skilled in the art will understand that time interval 406 can be dependent, at least in part, on the particular hardware and configuration of the analog processor, and the digital processor comprising the hybrid processor. The time taken by the digital processor to reprogram the analog processor and remove the applied preparatory biases may be different than shown in FIG. 4B. In some implementations, time interval 406 may range, for example, from 100 us to 200 μs. When the analog processor is a quantum processor, the digital processor may pause the evolution and retain the target classical spin state by keeping the energy barrier of the qubits high.

Additionally or in alternative, the hybrid processor may pause the evolution of the analog processor for a time interval longer than needed to reprogram the analog processor, thereby performing other operations, such as readout or post-processing, during time interval 406.

After time interval 406, the evolution of the analog processor resumes in a direction opposite the direction before time interval 406, i.e. backwards (also referred to in the present application as in a reverse direction). During this phase, the normalized evolution coefficient s decreases from 1 to a value s* at time t₃. The digital processor may determine the value of s* before the start of example evolution 400b, or during time interval 406.

Where the analog processor is a quantum processor, after time interval 406, the energy barriers of the qubits are lowered until an intermediate transverse field and/or tunneling energy is reached. The intermediate transverse field and/or tunneling energy may be determined by the digital processor.

After time t₃, the evolution of the analog processor is paused for a time interval 408 (between times t₃and t₄). Time interval 408 may be determined by the digital processor, either before the start of example evolution 400b or during time interval 406. In some implementations, time interval 408 may, for example, range from 1 us to several milliseconds.

A person skilled in the art will understand that the rate of change of the normalized evolution coefficient s between time t₂and time t₃may be the same as the rate of change between 0 and time t₁, or may be different. The digital processor may, for example, determine the rate of change of the normalized evolution coefficient.

After time interval 408, the evolution of the analog processor resumes in the same direction as the evolution from 0 to time t₁, i.e. the normalized evolution coefficient s increases from value s* to 1 until the analog processor reaches a classical spin state at time t₅. Where the analog processor is a quantum processor, the digital processor may raise the energy barriers of the qubits to reach a classical spin state. The classical spin state reached at time t₅may not be the same as the classical spin state reached at time t₁, given that the preparatory biases have been removed at time interval 406.

After time t₅, the digital processor may read out the classical spin state reached at t₅, and may perform post-processing.

In an alternative implementation, the hybrid processor performs post-processing on the obtained classical spin states at time interval 406 using classical methods. Therefore, the evolution of the analog processor is paused for a length of time necessary for the digital processor to perform the post-processing operations. An example of a classical post-processing method is Houdayer cluster moves, performed a predetermined number of times. Other classical post-processing methods can be used.

Alternatively, or in addition, post-processing may be used to improve samples obtained by the analog processor at time t₁. In an effort to improve the diversity of the samples obtained from the analog processor, the samples obtained at t₁can be post processed as described above and used as feedback to run the evolution of the analog processor one or more times. During the time interval 406, after the digital processor has completed the post-processing operation, the digital processor can apply preparatory biases to the analog processor using the post-processed samples as input to influence the evolution of the analog processor towards obtaining a more diverse set of samples (e.g., obtaining samples from regions in the energy landscape that had not been previously explored by the analog processor). At time t₂, the evolution of the processor resumes backwards (i.e., in reverse) as described above until the normalized evolution coefficient reaches value s* at t₃. As noted above, the samples obtained at t₅may not be the same as the samples obtained at t₁or the post-processed samples at t₁. After time t, the digital processor may read out the samples obtained by the analog processor.

FIG. 5 is a graph 500 of an example evolution 500 of an analog processor operating with a digital processor to form a hybrid processor according to the present systems, methods and apparatus, where the analog processor evolves backwards and forwards over time in the course of an annealing schedule. (Backwards evolution is also referred to as reverse annealing in the present application.) An analog processor may be a quantum processor comprising superconducting qubits and couplers. Vertical axis 502 represents the normalized evolution coefficient s* and the horizontal axis 504 represents the time of the evolution of the analog processor.

Before the start of example evolution 500, the digital processor may determine a set of normalized evolution coefficients as follows:

s
*

=

{

s
1
*

,

s
2
*

,

s
3
*

,
…

,

s
n
*

}

Example evolution 500 resembles example evolution 400b until time t₅, which is described above.

After time t₅, the digital processor may read out the state of the analog processor and/or perform post-processing. At time t₆, the evolution of the analog processor resumes, and the normalized evolution coefficient s decreases from 1 to value s₂* at time t₇. Time interval 510 is a time interval between t₅and t₆.

The value of s₂* may be different from the value of s₁* and, similarly, the rate of change of the normalized evolution coefficient s between t₆and t₇may be different from the rate of change of the normalized evolution coefficient s at other times during example evolution 500.

After a time interval 512, the evolution of the analog processor continues until the analog processor reaches a classical spin state at time t, and the normalized evolution coefficient s reaches a value of 1. As noted before with respect to example evolution 400b, the classical spin state reached at to may not be the same as the classical spin state reached at t₁and/or t₅.

After time t₉, the evolution of the analog processor is paused for a time interval 514, where time interval 514 may be determined by the digital processor. During time interval 514 the digital processor may read out the state of the analog processor and/or perform post processing.

After time t₁₀, the evolution of the analog processor resumes in a similar pattern, evolving to a predetermined value of the normalized evolution coefficient s, pausing for a time interval, and resuming until the analog processor reaches a classical spin state, at times t₁₁, t₁₂, and t₁₃, respectively.

Example evolution 500 may be used to study a particular problem dynamic, or to generate sample from the analog processor.

FIG. 6 illustrates a flow diagram of a computational method 600 using a hybrid computing system for evolving an analog processor where the analog processor evolves backwards and forwards over time over the course of an annealing schedule. The hybrid computing system comprises a digital processor and a quantum processor.

Computational method 600 comprises acts 602 to 630; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 600 starts at 602, for example in response to a call from another routine.

At 604, the digital processor determines a classical spin state for the analog processor. A classical spin state is a set of spin configurations as follows:

S
i

=

{

-
1

,

+
1

}

Computational method 600 will initially evolve the analog processor towards this classical spin state.

At 606, the digital processor receives an Ising problem to be solved via the analog processor. Such an Ising problem may be, for example, an optimization problem or a sampling problem.

At 608, the digital processor determines a set of preparatory biases that need to be applied to the elements of the analog processor so that the analog processor will evolve towards the classical spin state determined at 604. Where the analog processor is a quantum processor, the preparatory biases may be the flux biases applied to some or all of the qubits in the quantum processor. Preparatory biases can influence the evolution of the quantum processor towards the classical spin state determined at 604 so that the classical spin state is achieved with high fidelity (i.e., the probability of achieving the classical spin state is close to unity e.g. 0.9999).

At 610, the digital processor programs the analog processor with the Ising problem received at 606. The digital processor will program the h and/values of the Ising problem. Where the analog processor is a quantum processor, the digital processor will apply h and/values to the qubits and couplers of the quantum processor.

At 612, the digital processor programs the analog processor with the preparatory biases determined at 608. Where the analog processor is a quantum processor, the digital processor may load one or more pulses to the most significant digit of the qubits' flux bias DACs, in addition to the Ising problem bias term h. For example, two or three steps of the most significant digit of the qubits' flux bias DACs may be applied, so that qubits can be biased in the desired direction corresponding to the classical spin state determined at 604.

At 614, the analog processor evolves towards the classical spin state determined at 604. The rate of evolution at 614 may not be constant so that the evolution may be non-linear, e.g., ramping up at a certain stage or pausing before resuming towards the classical spin state.

At 616, the digital processor latches the state of analog processor for a first dwell time. Where the analog processor is a quantum processor, the qubits' energy barriers may be kept high for the first dwell time to retain the classical spin state. The digital processor may determine the first dwell time to be at least the time needed to reprogram the analog processor by removing the preparatory biases. In other implementations, the first dwell time may be longer.

At 618, the digital processor reprograms the analog processor to remove the preparatory biases, and the analog processor is programmed with the Ising problem received at 606. The time taken by this operation may depend on the particular configuration of the analog processor and the digital processor. Where the analog processor is a quantum processor, the digital processor can remove the one or more pulses to the most significant digit of the qubits' flux bias DACs that were applied at 612, leaving the bias term h of the Ising problem received at 606 such that the quantum processor is now programmed with the Ising problem only.

At 620, the digital processor determines evolution parameters including an intermediate tunneling energy and a second dwell time. The second dwell time can be independent from the analog processor programming time and may be different from the first dwell time.

At 622, the analog processor evolves in a backward direction until the intermediate tunneling energy is reached. Where the analog processor is a quantum processor, the qubits' energy barriers can be lowered to achieve the intermediate tunneling energy so that qubits in the quantum processor may not be in a classical spin state at 622.

In some implementations, one or more variables can be clamped in a classical spin state. Where the analog processor is a quantum processor, the clamped variables can each be represented by a respective one or more qubits, and the qubits representing the clamped variables can form a first subset of qubits of the quantum processor. At 622, the analog processor can evolve a second subset of qubits of the quantum processor, the second subset excluding qubits in the first subset of qubits (i.e., excluding qubits representing the clamped variables), in a backward direction until the intermediate tunneling energy is reached. The energy barriers of the second subset of qubits can be lowered to achieve the intermediate tunneling energy so that qubits in the second subset of qubits in the quantum processor may not be in a classical spin state at 622.

At 624, the digital processor pauses the analog processor for the second dwell time determined at 620. Where the analog processor is a quantum processor, the qubits' energy barriers may be kept at the intermediate tunneling energy level so that the evolution of the quantum processor is paused.

At 626, the analog processor evolves towards a classical spin state. Where the analog processor is a quantum processor, the digital processor raises the energy barrier of the qubits to evolve the quantum processor towards a classical spin state. The classical spin state reached at 626 may not be the same classical spin state reached at 614, owing to the removal of the preparatory biases at 618.

At 628, the digital processor reads out the state of the analog processor. During the read-out operation, the evolution of the analog processor is paused so that it maintains a classical spin state. Several methods may be employed for reading out the state of the analog processor, an example of a method and apparatus for reading out a state of a quantum processor are described in PCT Patent application PCT/US2016/031885.

At 630, computational method 600 ends, for example, until invoked again.

FIG. 7 shows a flow diagram of a computational method 700 using a hybrid computing system for evolving an analog processor over time, where the analog processor evolves backwards and forwards over the course of an annealing schedule. The hybrid computing system comprises a digital processor and a quantum processor.

Computational method 700 may be implemented as an extension of computational method 600 of FIG. 6 and comprises acts 702 to 732; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 700 starts at 702, for example in response to a call from another routine.

At 704, the digital processor determines a classical spin state configuration, as described above with reference to 604 of computational method 600.

At 706, the digital processor receives an Ising problem to be solved by the analog processor, as described above with reference to 606 of computational method 600.

At 708, the digital processor determines a set of preparatory biases, as described above with reference to 608 of computational method 600.

At 710, the digital processor programs the analog processor with the Ising problem, as described above with reference to 610 of computational method 600.

At 712, the digital processor programs the analog processor with the preparatory biases, as described above with reference to 612 of computational method 600.

At 714, the analog processor evolves towards the classical spin state, as described above with reference to 614 of computational method 600.

At 716, the digital processor latches the state of analog processor for a first dwell time, as described above with reference to 616 of computational method 600.

At 718, the digital processor reprograms the analog processor to remove the preparatory biases, as described above with reference to 618 of computational method 600.

At 720, the digital processor determines evolution parameters including an intermediate tunneling energy and a second dwell time, as described above with reference to 620 of computational method 600.

At 722, the analog processor evolves in a backward direction until the intermediate tunneling energy is reached, as described above with reference to 622 of computational method 600.

At 724 the digital processor pauses the analog processor for second dwell time, as described above with reference to 624 of computational method 600.

At 726, the analog processor evolves towards a classical spin state, as described above with reference to 626 of computational method 600.

At 728, the digital processor reads out the state of the analog processor.

At 730, the digital processor determines whether to iterate based on an exit condition. In response to an exit condition not being met, control proceeds to 712, and the digital processor performs a further iteration of acts 712 to 728. At 712, the digital processor programs the analog processor with preparatory biases. In response to the exit condition being met, control proceeds to 732. An exit condition may comprise iterating for defined number of times.

At 732, computational method 700 terminates, for example, until invoked again.

FIG. 8 is a flow diagram of a computational method 800 using a hybrid computing system for evolving an analog processor over time, where the analog processor iterates forwards and backwards over the course of an annealing schedule and where the digital processor does not reprogram the analog processor at each iteration.

Computational method 800 comprises acts 802 to 830; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 800 starts at 802, for example in response to a call from another routine.

At 804, the digital processor determines a classical spin state for the analog processor, as described above with reference to 604 of computational method 600 and/or 704 of computational method 700.

At 806, the digital processor receives an Ising problem to be solved by the analog processor. Such an Ising problem may be, for example, an optimization problem or a sampling problem. The digital processor also receives a set of normalized evolution coefficients:

s
*

=

{

s
1
*

,

s
2
*

,

s
3
*

,
…

,

s
n
*

}

- and a set of dwell times:

t
*

=

{

t
1
*

,

t
2
*

,

t
3
*

,
…

,

t
n
*

}

At 808, the digital processor determines a set of preparatory biases, as described above with reference to 608 of computational method 600 and/or 708 of method 700.

At 810, the digital processor programs the analog processor with the Ising problem as described above with reference to 610 of computational method 600 and/or 710 of computational method 700.

At 812, the digital processor programs the analog processor with the preparatory biases as described above with reference to 612 of computational method 600 and/or 712 of computational method 700.

At 814, the analog processor evolves towards the classical spin state, as described above with reference to 614 of computational method 600 and/or 714 of computational method 800.

At 816, the digital processor latches the state of the analog processor for a first dwell time, as described above with reference to 616 of computational method 600 and/or 716 of computational method 700.

At 818, the digital processor reprograms the analog processor to remove the preparatory biases, as described above with reference to 618 of computational method 600 and/or 718 of method 700.

At 820, the analog processor evolves backwards until the normalized evolution coefficient s reaches value s₁*, the first value of s in the set s* received by the digital processor at 806. Where the analog processor is a quantum processor, the energy barrier is lowered until the normalized evolution coefficient s reaches value s₁*. When s has value s₁*, qubits in the quantum processor may not be in a classical spin state.

At 822, the digital processor pauses the quantum processor for a dwell time t₁*, where t₁* is the first value in the set of dwell times t* received by the digital processor at 806.

At 824, the analog processor evolves towards a classical spin state. Where the analog processor is a quantum processor, the digital processor raises the energy barrier of the qubits to evolve the quantum processor towards a classical spin state. The classical spin state reached at 824 may not be the same classical spin state reached at 814.

At 826, the digital processor reads out the state of the analog processor.

At 828, the digital processor determines whether to iterate based on an exit condition. In response to an exit condition not being met, control proceeds to 820, and the digital processor iterates through acts 820 to 826 of method 800. At 820, the analog processor evolves backwards until the next value s* is reached (e.g., if in the previous iteration the backward anneal was paused at s₃* in the current iteration the backward anneal will pause at s₄*). Similarly, at 822, the digital processor latches the state of the analog processor for a dwell time corresponding to the next value of t* (e.g., if in the previous iteration the state of the analog computer was latched for dwell time t₃* in the current iteration the state of the analog computer will be latched for a dwell time t₄*).

In response to the exit condition being met, control proceeds to 830. An exit condition may comprise iterating for defined number of times. For example, computational method 800 may iterate n times, where n is the size of the sets s* and t*. At 830, computational method 800 terminates, for example, until invoked again.

Compared to computational method 700, the time taken to reprogram the processor with the Ising problem and the preparatory biases can be saved, or at least reduced, in computational method 800. Computational method 800 can have a faster execution cycle than that of computational method 700.

Computational methods 600, 700 and/or 800 may be used to operate a hybrid computer for studying problem dynamics, where an initial classical spin state is determined and a study of the escape rate out of the classical spin state at different points during an annealing schedule is desired. Alternatively, or additionally, computational methods 700 and/or 800 may be used for sampling. By repeating backward and forward annealing, the analog processor can explore neighborhoods close to the initial classical spin configuration. For example, computational methods 700 and/or 800 may be used for approximating Boltzmann sampling distributions.

Read out may be performed at each iteration of computational methods 700 and/or 800 (acts 726 and 826, respectively), or the state of the analog processor may be temporarily stored in a non-transitory memory storage device until the end of the iterations, and the digital processor may read out all the classical spin state at the end of computational methods 700 and/or 800, respectively. Where the analog processor is a quantum processor, a quantum flux parametron (QFP) shift register, with n QFPs per qubit, where n is the number of iterations of computational methods 700 and/or 800, may store all the classical spin states obtained during the execution of computational methods 700 and/or 800.

A person skilled in the may understand that the hybrid computing system may operate computational method 700 and 800 incrementally, wherein computational method 700 constitutes an outer loop and acts 820-826 of computational method 800 an inner loop.

FIG. 9 is a graph of an example evolution 900 of an analog processor, operating with a digital processor to form a hybrid processor according to the present systems, methods and apparatus, where the analog processor evolves forwards and backwards over several intervals. An analog processor may comprise a quantum processor. Vertical axis 902 represents the normalized evolution coefficient s and the horizontal axis 904 the time of the evolution of the analog processor.

Before the start of the evolution the digital processor may program a problem onto the analog processor. Where the analog processor is a quantum processor, the digital processor may, for example, assign bias and coupling strengths to some, or all, of the qubits and couplers of the quantum processor. The digital processor determines an annealing schedule for the analog processor (e.g., the digital processor may determine the rate of the anneal).

In example evolution 900, the normalized evolution coefficient s increases from 0 to a value s₁in time t₁. A person skilled in the art will understand that the rate of the evolution from 0 to t₁is shown in FIG. 9 for illustration purposes only and in other implementations the rate of the evolution of the analog processor from 0 to s₁may be faster of slower than illustrated. In addition, where the analog processor is a quantum processor, some of the qubits in the quantum processor may have a different annealing rate than other qubits or they may start annealing at a later time.

At time t₁the digital processor programs the analog processor with a first candidate annealing schedule. The first candidate annealing schedule may be the same as the initial annealing schedule determined by the digital processor before the start of the evolution.

At time t₁the evolution of the analog processor may be paused for a time necessary to program the candidate annealing schedule (not shown in FIG. 9) or for other purposes, before resuming until the normalized evolution coefficient s reaches value s₂at time t₂, where s₂>s₁. The values s₁and s₂, and/or t₁and t₂, may be determined by the digital processor before the start of the evolution and may be determined, at least in part, by the class of problem that is to be programmed into the analog processor.

At time t₂the evolution of the analog processor proceeds in an opposite direction (i.e., backwards) with respect to the direction of the evolution up to time t₂. At time t₃the normalized evolution coefficient s decreases to value s₁.

At time t₃the digital processor programs the analog processor with a second candidate annealing schedule that may be different from the first candidate annealing schedule. The evolution of the analog processor may be paused for the time needed to program the second candidate annealing schedule into the analog processor.

After time t₃the evolution of the analog processor proceeds in the first direction (i.e., forward) until a time ta when the normalized evolution coefficient s reaches value s₂again, before proceeding in the opposite direction (i.e., backwards) until the normalized evolution coefficient reaches value s₁at time t₅.

At times t₂and t₄, the digital processor may readout the spin configurations of the analog processor.

While in FIG. 9 the evolution of the analog processor is shown to move forward and backwards between the values s₁and s₂two times, a person skilled in the art will understand that the analog processor may evolve between 1 and s₂more than two times, or only once.

Similarly, the evolution of the analog processor proceeds forwards and backwards between s₂and s₃, and successively between values of the normalized evolution coefficient s₃and 1.

Although in FIG. 9 the evolution of the analog processor is shown to proceeds forwards and backwards between three values s₁-s₃of the normalized evolution coefficient s, a person skilled in the art will understand that the evolution of the analog processor may proceed as described between less than three or more than three values of the normalized evolution coefficient s.

At time t₁₄, after intervening times t₆through t₁₃, the analog processor can have tried a number of candidate annealing schedules between intervals of the normalized evolution coefficient s to attempt to find an optimal annealing schedule for each interval. Depending on the problem, or problem class, to be solved by the analog processor, a specific annealing schedule may be more suited than others to find, for example, a more diverse set of samples, a solution with a lower energy, or a solution that requires less post-processing.

FIG. 10 is a flow diagram illustrating a computational method 1000 using a hybrid computing system for evolving an analog processor over time, where the analog processor evolves forwards and backwards over intervals of the normalized evolution coefficient s in an attempt to determine a more suitable annealing schedule for each interval.

Computational method 1000 comprises acts 1002 to 1020; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 1000 starts at 1002, for example in response to a call from another routine.

At 1004 the digital processor initializes a counter i to an initial value i=0 and determines the number of intervals of the normalized evolution coefficient s for which method 1000 should run. The number of intervals can determine the value of the counter i. The digital processor may also determine a set of candidate annealing schedules for each interval of s. Alternatively, a set of candidate schedules may be determined by a separate process or routine, and passed to the digital processor as an input to computational method 1000, or the set of candidate schedules for each interval of i+1 can be determined after the candidate schedule for interval i has been computed in an iteration of computational method 1000.

At 1006 the digital processor programs the analog processor with one of the candidate annealing schedules for the interval [s_i, s_i+1].

At 1008 the analog processor starts an evolution in a first direction following the candidate annealing schedule programmed at 1006 until the normalized evolution coefficient reaches value s_i+1. Typically, the first direction is a forward direction (i.e. towards s=1).

At 1010 the digital processor reads out the state of the analog processor. The digital processor may store this information in a memory element for future calculation and/comparison. In some implementations, the analog processor may need to evolve until s=1 before the digital processor can read out the state of the analog processor. In evolving until s=1, the analog processor may evolve faster than to the evolution at 1008. In some cases, the analog processor may follow a ramp to s=1.

At 1012 the digital processor determines whether an exit condition has been met. In one implementation, the exit condition is completion of a defined number of iterations, the number of iterations corresponding to the number of candidate annealing schedules for the interval [s_i, s_i+1]. In this implementation, the exit condition ensures that the candidate annealing schedules have been tried. If the exit condition has been met, control passes to 1016, otherwise to 1014.

At 1014, evolution of the analog processor proceeds in the opposite direction (typically in the backwards or reverse direction i.e., away from s=1) until the normalized evolution coefficient decreases to value s_iagain. After 1014, control returns to 1006, where the digital processor programs the analog processor with a different one of the candidate schedules determined at 1004.

At 1016, the digital processor determines whether for the current value of the counter i the condition s_i+1=1 is met. If the condition is met, evolution of the analog processor has reached the end, and computational method 1000 has iterated over the previous intervals of s, and method 1000 proceeds to 1020. Computational method 1000 terminates at 1020, until it is invoked again, for example. Alternatively, before terminating execution of computational method 1000, the digital processor may determine a more suitable annealing schedule for each interval, based at least in part on the readout information collected at 1010, and program the analog processor to evolve according to the more suitable annealing schedule for each interval.

If the condition is not met, control passes to 1018. At 1018, the digital processor increments the value of the counter i to i+1, progressing execution of computational method 1000 to the next interval of s. Control then passes to 1006, where the digital processor programs the analog processor with one of the candidate annealing schedules for the interval [s_i, s_i+1] determined at 1004.

An analog processor may solve a computational problem involving more than one variable. Where the analog processor is a quantum processor, the variables of the problems may be represented by qubits.

Depending on the problem and the topology of the quantum processor, one or more variables in the problem may be represented in the quantum processor by more than one qubit. For example, one variable may be represented by a group of two or more qubits that are influenced to behave as a single qubit by programmable couplers. Such groups of qubits are commonly referred to as chains. These chains are distinct from the QMC chains and MCMC chains described above. The qubits in a chain can take the same spin as each other at the end of the evolution of the quantum processor (either spin up or spin down).

The digital processor may assign a coupling strength to the programmable couplers so that the qubits in the chain behave as a single qubit. The stronger the chain, the more likely it is that the qubits in the chain can behave as a single qubit. The coupling strength, or chain strength, may vary over the evolution of the quantum processor. For example, the chain strength may be defined as a ratio of the coupling strength of qubits in the chain to the strongest logical graph coupling.

FIG. 11 is a graph of an exemplary variation of chain strength over the course of an evolution of a hybrid computing system comprising a digital processor in communication with an analog processor. In particular, FIG. 11 shows an exemplary variation of chain strengths 1102a, 1102b, 1102c, and 1102d (collectively 1102) of four variables over the course of an evolution of an analog processor. The vertical axis 1104 represents the chain strengths of the four variables and the horizontal axis 1106 represents the time of the evolution of the analog processor. One or more variables may be represented in a quantum processor by more than one qubit, so that the chain strength of one variable may be the combination of the coupler strengths of the couplers connecting the qubits in the chain.

In FIG. 11, chain strength 1102a remains approximately constant over the course of the evolution of the analog processor, while chain strengths 1104b, 1104c, and 1104d vary. A person skilled in the art will understand that the analog processor may be representing a computational problem with more than four variables, or with less than four variables, and the chain strength of each variable may be approximately constant, or vary over the course of the evolution, or be approximately constant for a phase of the evolution and vary over another phase of the evolution.

The digital processor in the hybrid computing system programs the chain strength of each chain at the beginning of the evolution, thus setting the chain strength to an initial value. However, during the evolution, some of the chain strengths may vary, depending on, for example, external influence or noise in the analog processor. Therefore, the initial value of the chain strength may not be an optimal value, or a preferred value, for throughout the evolution.

To compensate for variation in chain strength, the digital processor may pause the evolution of the analog processor at predetermined times to read out the chain strength of the variables. In FIG. 11, the digital processor may read out the chain strength of the four variables at time intervals of Δt, at times t₁to t₈.

While in FIG. 11 the digital processor reads out the values of the chain strengths at regular time intervals, a person skilled in the art will understand that in a different implementation, the digital processor may read out the values of the chain strengths at irregular (or unequal) intervals.

FIG. 12 is a flow diagram of a computational method 1200 using a hybrid computing system for evolving an analog processor where the chain strengths of the variables in the analog processor changes over the course of the evolution and where the digital processor determines a more optimal, or preferred, chain strength for each problem variable, and for each interval, based on reading out the chain strengths during the evolutions. The analog processor may be a quantum processor.

Computational method 1200 comprises acts 1202 to 1220; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 1200 starts at 1202, for example in response to a call from another routine.

At 1204, the digital processor initializes a counter i to an initial value i=0. The digital processor determines a number of intervals of the normalized evolution coefficient s for which method 1200 should run. The number of intervals can determine the value of the counter i. At the same time, the digital processor may program other elements of the analog processor in order to prepare the analog processor for solving a computational problem. The digital processor also determines, based at least in part on the problem to be solved, a set of candidate chain strengths for each interval. Alternatively, the set of chain strengths may be determined by a separate routine, and passed as an input to computational method 1200, or the set of candidate chain strengths for each interval of i+1 can be determined after the candidate chain strengths for interval i have been computed via an iteration of computational method 1200.

The chain strength for each chain can depend, at least in part, on the problem to be solved by the analog processor and/or, in the case of a quantum processor, on the number of qubits comprising each variable and/or other characteristics of the quantum processor.

At 1206 the digital processor programs the analog processor with one of the candidate chain strengths from the set of candidate chain strengths. The digital processor may also program other parameters of the analog processor, as necessary.

At 1208, the evolution of the analog processor starts and continues until a value of the normalized evolution coefficient s_i+1.

At 1210, the digital processor reads out the values of the chain strengths of the variables represented by chains in the analog processor. At the same time, the digital processor may read out other values of other elements of the analog processor. To do so, the evolution of the analog processor may pause for a time necessary to carry out the readout operation. Additionally, or in alternative, the analog processor may evolve at a faster rate, or ramp up the evolution, until the normalized evolution coefficient reaches value 1 before carrying out the readout operation.

At 1212, the digital processor determines whether an exit condition has been met. If the exit condition has been met, control passes to 1216, otherwise control passes to 1214. An exit condition may be, for example, the execution of a number of iterations, where the number of iterations corresponds to the number of candidate chain strengths for each interval of the normalized evolution coefficient. An alternative exit condition may be reaching a threshold of a performance measure.

At 1214, the evolution of the analog processor proceeds in an opposite direction (i.e., in a backwards or reverse direction—the reverse evolution also referred to in the present application as a reverse anneal) until the value of the normalized evolution coefficient s decreases to s_i. Control then passes to 1206 where the digital processor programs the analog processor with another one of the candidate chain strengths from the set of candidate chain strengths. The digital processor may also program other parameters of the analog processor, as desired. Alternatively, or in addition, the digital processor may program the chain strengths based on a combination of another one of the candidate chain strengths and the readout at 1208.

At 1216, the digital processor determines whether the value s_i+1=1. In this case, all the analog processor will have tried all the candidate chain strengths for all the intervals of the normalized evolution coefficient i. Upon meeting the condition control passes to 1220, otherwise to 1218.

At 1218, the digital processor increases the value of the counter i to i+1, progressing the execution of computational method 1200 to the next interval of s. Control then passes to 1206, where the digital processor programs a candidate chain strength for the next interval of s.

At 1220, computational method 1200 terminates, until it is invoked again. Alternatively, before terminating execution of computational method 1200, the digital processor may determine a more suitable chain strength for each interval, based at least in part on the readout information collected at 1208, and program the analog processor to evolve according to the more suitable chain strength for each interval of s.

FIG. 13 is a flow diagram of a computational method 1300 using a hybrid computing system for evolving an analog processor, where the analog processor evolves backwards and forwards to mitigate the effect of broken chains.

When a problem requires the use of chains to be programmed into the specific topology of the quantum processor, it may occur that at the end of the evolution of the quantum processor one or more chains contains qubits whose spin do not agree, e.g., some qubits in a single chain have spin up while other qubits in the same chains have spin down. In this case the chain is said to be broken.

Solutions containing broken chains may not be meaningful and may require post-processing. Additionally or in alternative, repeating solving the same problem with the quantum processor may help mitigate the effect of the broken chains.

Computational method 1300 attempts to mitigate the effect of broken chains in a solution by evolving only the broken chains in a quantum processor backwards and then evolving them forward.

Computational method 1300 comprises acts 1302 to 1318; however, a person skilled in the art will understand that the number of acts is exemplary and, in some implementations, certain acts may be omitted, further acts may be added, and/or the order of the acts may be changed.

Computational method 1300 starts at 1302, for example in response to a call from another routine.

At 1304, the analog processor solves a problem by evolving and reaching a solution. The digital processor may program the problem into the analog processor by programming the parameters of the analog processors, including chain strength.

At 1306, the digital processor reads out the state of the analog processor. Where the analog processor is a quantum processor, the digital processor reads out the spin configuration of the qubits.

At 1308, the digital processor determines if there are any broken chains in the solution read out at 1306. Where the analog processor is a quantum processor, the digital processor checks if the spin configurations of the qubits within a chain agree with one another. The digital processor can check one or more of the chains. If there are no broken chains in the solution read out at 1306, control passes to 1310, otherwise to 1312.

At 1312, the digital processor sets or programs the state of the analog processor so that unbroken chains are held fixed. Where the analog processor is a quantum processor, the digital processor can program the quantum processor by setting the spin configurations of qubits not belonging to a broken chain as fixed.

At 1314, the evolution of the analog processor proceeds backwards so that the value of the normalized evolution coefficient s decreases. Given that the state of the non-broken chains has been set at 1312, only a portion of the analog processor will evolve backwards. The digital processor will determine when to pause the backwards anneal of the analog processor, trying to pause it at a time when the chains were unbroken. In order to do so, the digital processor may have to read out the state of the analog processor one or more times at various points during the backwards (reverse) anneal.

At 1316, evolution of the analog processor proceeds forward, until the end of the evolution, when the normalized evolution coefficient s reaches a value of 1. Given that unbroken chains were held fixed at 1312, only the portion of the processor that had broken chains at 1308 is evolving forward.

At 1318, the digital processor reads out the state of the analog processor. After 1318, control passes back to 1308, where the digital processor determines if there are broken chains in the solution read out at 1318. Method 1300 iterates until there are no broken chains in the solution.

At 1310, computational method 1300 terminates, until, for example, it is invoked again.

Alternatively, computational method 1300 may be implemented without evolving the analog processor backwards. At 1314, the analog processor, instead of evolving backwards, may start a new evolution cycle for the broken chains only, where the unbroken chains are set fixed at 1312.

As described above, the digital processor can, at each iteration, check if there are broken chains in a solution obtained by the analog processor, and repeat evolving the broken chains until a result is produced with no broken chains. Alternatively, the digital processor may determine to stop the iterations based on the completion of a defined number of iterations.

Other methods exist for post-processing samples obtained with an analog processor. Some post processing techniques are described in US Patent Publication No US2015363708, U.S. Pat. Nos. 7,307,275, 8,244,650, 8,494,993 and US Patent Publication No 20160071021.

Some approaches to post-processing are classical, i.e., they use a digital processor to post-process samples from an analog processor. As described in the present application, other approaches use a hybrid computing system, the hybrid computing system comprising an analog processor and a digital processor, where the post-processing operation is performed on the analog processor.