QUANTUM HARDWARE CHARACTERIZED BY PROGRAMMABLE BOSE-HUBBARD HAMILTONIANS

Abstract

An apparatus includes a first group of superconducting cavities and a second group of superconducting cavities, each of which is configured to receive multiple photons. The apparatus includes couplers, where each coupler couples one superconducting cavity from the first group with one cavity from the second group such that the photons in the coupled superconducting cavities interact. A first superconducting cavity of the first group is connected to a second superconducting cavity of the second group, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities. The first superconducting cavity is coupled to at least one other superconducting cavity of the first group to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to at least one other superconducting cavity of the second group to which the first superconducting cavities are coupled.

Description

BACKGROUND

The present specification relates to quantum hardware characterized by programmable Bose-Hubbard Hamiltonians.

SUMMARY

In a computational paradigm of this specification, quantum information is represented by multimode quantum hardware, the dynamics of which can be characterized and controlled by a programmable many-body quantum Hamiltonian. The multimode quantum hardware can be programmed as, for example, a quantum processor for certain machine learning problems. Examples of the quantum hardware include neutral atoms on optical lattices, photonic integrated circuits, or superconducting cavity quantum electrodynamics (QED) circuits, and the Hamiltonians characterizing such quantum hardware include dissipative or non-dissipative Bose-Hubbard Hamiltonians.

The solution to a machine optimization problem can be encoded into an energy spectrum of a Bose-Hubbard quantum Hamiltonian. For example, the solution is encoded in the ground state of the Hamiltonian. Through an annealing process in which the Hamiltonian evolves from an initial Hamiltonian into a problem Hamiltonian, the energy spectrum or the ground state of the Hamiltonian for solving the problem can be obtained without diagonalizing the Hamiltonian. The annealing process may not require tensor product structure of conventional qubits or rotations and measurements of conventional local single qubits. In addition, quantum noise or dechoerence can act as a recourse to drive the non-equilibrium quantum dynamics into a non-trivial steady state. The quantum hardware can be used to solve a richer set of problems as compared to quantum hardware represented by an Ising Hamiltonian. Furthermore, instead of the binary representations provided by the Ising Hamiltonians, constraint functions of problems to be solved can have a digital representation according to the density of states in Cavity QED modes.

In general, in some aspects, the subject matter of the present disclosure can be embodied in apparatuses that include: a first group of superconducting cavities each configured to receive multiple photons; a second group of superconducting cavities each configured to receive multiple photons; and multiple couplers, in which each coupler couples one superconducting cavity from the first group of superconducting cavities with one superconducting cavity from the second group of superconducting cavities such that the photons in the coupled superconducting cavities interact, and in which a first superconducting cavity of the first group of superconducting cavities is connected to a second superconducting cavity of the second group of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first group of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one or more of the other superconducting cavities of the second group of superconducting cavities to which the first superconducting cavities are coupled.

Various implementations of the apparatuses are possible. For example, in some implementations, each coupler is configured to annihilate a photon in one superconducting cavity and create a photon in a different superconducting cavity.

In some implementations, at least one of the couplers includes a Josephson junction.

In some implementations, a Hamiltonian characterizing the apparatus is: Σ_ih_in_i+Σ_i,jt_ij(α_i^†α_j+h.c.)+Σ_iU_in_i(n_i−1), in which n_i is a particle number operator and denotes occupation number of a cavity mode i, α_i^† is a creation operator that creates a photon in cavity mode i, α_i is an annihilation operator that annihilates a photon in cavity mode j, h_i corresponds to a site disorder, U_i corresponds to an on-site interaction, t_i,j are the hopping matrix elements, and h.c. is hermitian conjugate. In some implementations, the multiple couplers are trained to produce an output desired probability density function at a subsystem of interest at an equilibrium state of the apparatus. In some implementations, the apparatuses are trained as Quantum Boltzmann machines.

In some implementations, wherein a Hamiltonian characterizing the apparatus is: Σ_ih_in_i+Σ_i,jt_ij(α_i^†α_j+h.c.)+Σ_iU_in_i(n_i−1)+Σ_i,jU_ijn_in_j, in which n_i is a particle number operator and denotes occupation number of a cavity mode i, α_i^† is a creation operator that creates a photon in cavity mode i, α_i is an annihilation operator that annihilates a photon in cavity mode j, h_i corresponds to a site disorder, U_i corresponds to an on-site interaction, t_i,j are the hopping matrix elements, and h.c. is hermitian conjugate. The apparatuses can be operable to evolve adiabatically to a ground state of a problem Hamiltonian H_p=Σ_i h_in_i+Σ_iU_in_i(n_i+1)+Σ_i,jU_ijn_in_j. The apparatus can be operable to evolve adiabatically from a Mott-insulator state to a superfluid state, in which an initial Hamiltonian of the apparatus is H_i=Σ_i,jt_ij(α_i^†α_j+h.c.). The apparatus can be operable to evolve adiabatically from a Mott-insulator state to a ground state of a problem Hamiltonian H_p=Σ_ih_in_i+Σ_iU_in_i(n_i−1)+Σ_i,jU_ijn_in_j, in which an initial Hamiltonian of the apparatus is H_i=Σ_i,jt_ij(α_i^†+h.c.).

In some implementations, the apparatus is configured to respond to an external field ε(t) and a Hamiltonian characterizing the apparatus in the external field is: Σ_ih_in_i+Σ_i,jt_ij(α_i^†α_j+h.c.)+Σ_iU_in_i(n_i−1)+Σ_i[ε(t)α_i^†+ε(t)*α_i]+H_SB, in which H_SB=Σ_iΣ_υ[κ_i,υ(α_ib_υ^†+α_i^†b_υ)]+λ_i,υα_i^†α_i(b_υ+b_υ^†), and in which n_i is a particle number operator, ε(t) is a slowly-varying envelope of an externally applied field to compensate for photon loss, H_SB is a Hamiltonian of the interaction between the apparatus and a background bath in which the apparatus is located, b_υ and b_υ^† are annihilation and creation operators for a bosonic background bath environment, κ_i,υ is a strength of apparatus-bath interactions corresponding to exchange of energy, h_i corresponds to a site disorder, U_i corresponds to an on-site interaction, t_i,j are the hopping matrix elements, and λ_i,υ corresponds to a strength of local photon occupation fluctuations due to exchange of phase with the bath. The apparatus can be operable to be dissipatively-driven to a ground state of a problem Hamiltonian.

In some implementations, at least one cavity is a 2D cavity. For example, each cavity can be a 2D cavity.

In some implementations, at least one cavity is a 3D cavity. For example, each cavity can be a 3D cavity.

In some implementations, each superconducting cavity in the first group of superconducting cavities is connected to a superconducting cavity in the second group of superconducting cavities.

In general, in other aspects, the subject matter of the present disclosure can be embodied in methods that include providing an apparatus having: a first group of superconducting cavities each configured to receive multiple photons; a second group of superconducting cavities each configured to receive multiple photons; and multiple couplers, in which each coupler couples one superconducting cavity from the first group of superconducting cavities with one superconducting cavity from the second group of superconducting cavities such that the photons in the coupled superconducting cavities interact, and in which a first superconducting cavity of the first group of superconducting cavities is connected to a second superconducting cavity of the second group of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first group of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one or more of the other superconducting cavities of the second group of superconducting cavities to which the first superconducting cavities are coupled. The apparatus can be provided in an initial Mott-insulated state. The methods can further include causing a quantum phase transition of the apparatus from the initial Mott-insulator state to a superfluid sate; and adiabatically guiding the apparatus to a problem Hamiltonian.

Various implementations of the methods are possible. For example, in some implementations, the methods can further include causing a quantum phase transition of the apparatus from the superfluid state to a final Mott-insulator state and reading the state of each superconducting cavity in the apparatus. The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of an example structure of quantum hardware.

FIG. 2A is an example of a selected connection in quantum hardware.

FIG. 2B is an example of a full connection in quantum hardware.

FIG. 3 is a flow diagram of an example process for encoding a problem in a Hamiltonian containing density-density interactions and programming quantum hardware.

FIG. 4 is a flow diagram of an example process for encoding a problem in a dissipative-driven Hamiltonian and programming quantum hardware.

DETAILED DESCRIPTION

FIG. 1 is a schematic of an example structure of quantum hardware 100 that can be characterized by programmable Bose-Hubbard Hamiltonians. The quantum hardware 100 includes QED cavities 104 arranged in columns 110, and lines 112. At least some pairs of the QED cavities, such as cavities 110 and 112, are coupled to each other through coupler 106. The QED cavities can be superconducting waveguide cavities restricted in dimensionality, e.g., to 1D, 2D or 3D. The couplers 106 can be inductive couplers, and the hardware can be configured with resistors and inductors. The couplers 106 can be Josephson couplers, and in an example, a Josephson coupler is constructed by connecting two superconducting elements separated by an insulator and a capacitance in parallel.

The cavities contain photons in optical modes 102. The cavities can receive a variable amount of photons when the quantum hardware is initialized, or during the use of the quantum hardware. A coupler 106 between two cavities allows the photons of the two cavities to interact with each other. For example, the coupler can create or annihilate photons in a cavity, or move photons between cavities. Each cavity in the hardware 100 can be used as a logical computation unit. The number of photons in a cavity mode of the cavity can be read using photon detectors 108.

In some implementations, the quantum hardware 100 includes a fully connected network of superconducting cavities 104. In this network, each cavity is coupled with all other cavities through couplers 106. In other implementations, selected pairs of cavities are coupled with each other. The selection can be made based on the need for the quantum computation and the physical confinement of the hardware.

FIG. 2A is an example of a selected connection in quantum hardware. The hardware includes cavities A, B, C, D, E, F, G, and H, and pairs of cavities are coupled through couplers 208. Cavities “A” 202 and “E” 204 are selected to be connected through a connection 206, so that effectively, they become the same cavity. That is, the connection can be considered an extension of the cavity QED mode. Without the connection 206, the cavity “A” is coupled to cavities “E”, “F”, “G”, and “H,” but not to cavities B, C, and D.

Effectively, in this example, cavity “A” is coupled to all other cavities of the hardware. However, cavities “B”-“H” are only coupled to selected cavities of the hardware. To increase the number of cavities each cavity is coupled to, additional connections similar to the connection 206 can be added. The total amount of interaction between cavities in the hardware can be increased.

An example of a fully connected network is shown in FIG. 2B. FIG. 2B includes connections between “E” and “A” 212, “F” and “B” 214, “G” and “C” 216, and “H” and “D” 218. The network of FIG. 2B therefore allows for each cavity to interact with all other cavities.

The hardware of FIGS. 1, 2A, and 2B can be characterized by a Bose-Hubbard Hamiltonian:

$H=\sum _ih_in_i+\sum _{i,j}t_{\mathrm{ij}}\left(a_i^{†}a_j+h.c.\right)+\sum _iU_in_i\left(n_i-1\right)$

where n_i is the particle number operator and denotes the occupation number of a cavity mode i, α_i^† is a creation operator that creates a photon in cavity mode i, α_j is an annihilation operator that annihilates a photon in cavity mode j, h_i corresponds to a site disorder, U_i corresponds to an on-site interaction, are the hopping matrix elements, and h.c. is the hermitian conjugate.

The hardware of FIGS. 1, 2A, and 2B, characterized by the Bose-Hubbard Hamiltonian above, can be used to determine solutions to problems by training the hardware as a Quantum Boltzmann Machine for probabilistic inference on Markov Random Fields. For example, a problem can be defined by a set of observables y_i, e.g., photon occupation number at a cavity of the hardware, and a goal is to infer underlying correlations among a set of hidden variables x_i. Assuming statistical independence among various pairs of y_i and x_i, the joint probability distribution would be

$p\left(\left\{x_i\right\},\left\{y_i\right\}\right)=1/Z\prod _{i,j}{\theta }_{i,j}\left(x_i,x_j\right)\prod _i{\alpha }_i\left(x_i,y_i\right),$

where Z is the partition function (which, for a given system with a fixed energy function or a given Hamiltonian, is constant), θ_i,j(x_i, x_j) is a pairwise correlation, and α_i(x_i, y_i) is the statistical dependency between a given pair of y_i and x_i.

In training the hardware, certain cavity modes can act as the visible/observable input nodes of the Markov Random Field and can be used to train one or more of the Josephson couplers, which connect the hidden nodes x_i, to reproduce certain probability distribution of outcomes at the output visible nodes y_i.

For example, the training can be such that the delocalized energy ground state of the Bose-Hubbard model for each input state can have a probability distribution over the computational, i.e., localized, basis that resembles the output probability distribution function (PDF) of the training example, i.e., p({x_j}, {y_i}). Thus, the thermalized state of the hardware trained as a Quantum Boltzmann Machine can be sampled to provide a probabilistic inference on the test data according to the Boltzmann distribution function.

For example, an energy function can be defined:

$E\left(\left\{x_i\right\},\left\{y_i\right\}\right)=-\sum _{i,j}t_{i,j}\left(x_i,x_j\right)-\sum _ih_i(x_i,y_{i)}$

where the nonlocal pairwise interactions t_i,j(x_i, x_j)=θ_i,j(x_i, x_j), and disordered local fields h_i(x_i, y_i)=α_i(x_i, y_i).

The Boltzmann distribution of the above energy function is then:

$p\left(\left\{x_i\right\},\left\{y_i\right\}\right)=\frac{1}{Z}{}^{-\frac{E\left(\left\{x_i\right\},\left\{y_i\right\}\right)}{T}},$

where Z is the partition function.

In some other implementations, the quantum hardware of FIGS. 1, 2A, and 2B can be engineered or controlled to allow an additional type of coupling between the coupled cavities characterized by density-density interactions. With density-density interactions, an additional term can be added to the Bose-Hubbard Hamiltonian:

$\sum _{i,j}U_{\mathrm{ij}}n_in_j,$

The addition of the density-density interaction term to the Bose-Hubbard Hamiltonian can allow construction of a problem Hamiltonian in which the solution of a wide variety of problems can be encoded. For example, constraint functions of problems can have a digital representation according to the density of states in cavity QED modes with density-density interactions.

To account for photon loss in the above Hamiltonian, in some implementations the hardware is driven with additional fields to compensate for the loss.

To provide the density-density interaction term in the Hamiltonian, the quantum hardware can be engineered (an example process of using the additionally engineered hardware is shown in FIG. 3) or by controlling the hardware using a dissipative-driven method (an example process of using the controlled hardware is shown in FIG. 4).

Using the quantum hardware of FIG. 1 as an example, the quantum hardware can additionally be engineered to include, e.g., Kerr non-linearity with Josephson Junction couplers, the Stark effect, or continuous-time C-phase gates between cavities.

The modified Hamiltonian characterizing the additionally engineered hardware is therefore:

$H_{\mathrm{total}}=\sum _ih_in_i+\sum _{i,j}t_{\mathrm{ij}}\left(a_i^{†}a_j+h.c.\right)+\sum _iU_in_i\left(n_i-1\right)+\sum _{i,j}U_{\mathrm{ij}}n_in_j,$

where the final term is the density-density interactions between cavities i and j.

In use for adiabatic computation, a time dependent Hamiltonian can be represented as:

H_total=(1−s)H_i+sH_p,

where s is a control parameter and can be a linear function of time, H_i is the initial Hamiltonian:

$H_i=\sum _{i,j}t_{\mathrm{ij}}\left(a_i^{†}a_j+h.c.\right),$

and H_p is the problem Hamiltonian into which the selected problem is encoded:

H_p=Σ_ih_in_i+Σ_iU_in_i(n_i−1)+Σ_i,jU_ijn_in_j.

When s=0, the hardware is placed into an initial ground state that is known. The hardware is then quais-adiabatically guided to s=1, moving the hardware to the ground state of the Hamiltonian encoded by the problem.

FIG. 3 is a flow diagram of an example process 300 for encoding a problem in a Hamiltonian containing density-density interactions and programming quantum hardware.

In solving a given problem, e.g., an optimization problem, a problem modeled as a Markov Random Field, or an NP-Hard problem, the hardware undergoes a quantum phase transition from a Mott-insulator state to a superfluid state (step 302). The hardware is initially in an insulated state with no phase coherence, and with localized wavefunctions only. The many-body state is therefore a product of local Fock states for each cavity in the hardware:

|ψ_MI=Π_iα_i^N|0,

where N is the number of photons, and i is the cavity mode.

The hardware undergoes a quantum phase transition to a superfluid state so that the wavefunctions are spread out over the entire hardware:

${\psi }_{\mathrm{SF}}〉=\sum _ia_i^N0〉$

The hardware is adiabatically guided to a problem Hamiltonian (step 304). That is, the hardware is moved from the s=0 state, to the s=1 state as explained above.

At the end of the annealing process, the hardware transitions from a superfluid state to a non-trivial Mott-insulator state that can capture the solution to the problem (step 306).

The quantum state of the entire hardware is read out (step 308) and can be processed by a classical computer to provide solutions to the given problem. For example, the state of each cavity is determined by the photon occupation number of each cavity mode. The process 300 can be repeated multiple times for the given problem to provide solutions with a statistical distribution.

Alternatively, using the quantum hardware of FIG. 1 as an example, the dynamical effects of density-density interactions can be achieved by an interplay of the Bose-Hubbard Hamiltonian with cavity photon number fluctuations induced by an auxiliary external field. The combination of the hardware and the auxiliary external field is called a dissipative-driven hardware, and the Hamiltonian describing the dissipative-driven hardware is:

$H_{\mathrm{BH}}=\sum _ih_in_i+\sum _{i,j}t_{\mathrm{ij}}\left(a_i^{†}a_j+h.c.\right)+\sum _iU_in_i\left(n_i-1\right)+\sum _i\left[\varepsilon \left(t\right)a_i^{†}+{\varepsilon \left(t\right)}^{*}a_i\right]+H_{\mathrm{SB}}$

where ε(t) is a slowly-varying envelope of an externally applied field to compensate for photon loss, and H_SB is the Hamiltonian of the interaction between the hardware and the background bath in which the hardware is located:

$H_{\mathrm{SB}}=\sum _i\sum _v\left[{\kappa }_{i,v}\left(a_ib_v^{†}+a_i^{†}b_v\right)+{\lambda }_{i,v}a_i^{†}a_i\left(b_v+b_v^{†}\right)\right],$

where b_υ, and b_υ^† are annihilation and creation operators for the bosonic background bath environment, κ_i,υ is the strength of hardware-bath interactions corresponding to the exchange of energy, h_i corresponds to a site disorder, U_i corresponds to an on-site interaction, t_i,j are the hopping matrix elements, and λ_i,υ corresponds to the strength of local photon occupation fluctuations due to exchange of phase with the bath.

Using the dissipative-driven hardware, a solution to a problem can be determined without adiabatically guiding the hardware to the ground state of a problem Hamiltonian as in the process 300 of FIG. 3. The dissipative-driven hardware is eventually dominated by dissipative dynamics, defining a non-trivial steady state in which the solution to a problem is encoded.

FIG. 4 is a flow diagram of an example process 400 for encoding a problem in a dissipative-driven Hamiltonian and programming quantum hardware.

The hardware is programmed for a problem to be solved (step 402). In some implementations the problem is an optimization problem or an inference task and is mapped to a Markov Random Field. For example, a problem can be defined by a set of observables y_i, e.g., photon occupation number at a cavity of the hardware, and the goal is to infer underlying correlations among a set of hidden variables x_i. Assuming statistical independence among pair y_i and x_i, the joint probability distribution would be:

$p\left(\left\{x_i\right\},\left\{y_i\right\}\right)=1/Z\prod _{i,j}{\theta }_{i,j}\left(x_i,x_j\right)\prod _i{\alpha }_i\left(x_i,y_i\right),$

where Z is a normalization constant, θ_i,j(x_i, x_j) is a pairwise correlation, and α_i(x_i, y_i) is the statistical dependency between a given pair of y_i and x_i.

In many classes of machine learning problems, e.g., computer vision, image processing, and medical diagnosis, the goal of the problems is to compute marginal probabilities:

$p\left(x_N\right)=\sum _{x1}\sum _{x2}\dots \sum _{x_{N-1}}p\left(\left\{x_i\right\},\left\{y_i\right\}\right)$

Using a density-matrix formulation, the marginal probabilities can be computed from the dynamics of the dissipative-driven hardware in a quantum trajectory picture:

$\frac{\rho }{t}=-i\left[H_{\mathrm{BH}}+H_{\mathrm{LS}}+H_{\mathrm{decoh}},\rho \right]+\sum _{i,i^{\prime },j,j^{\prime }}{\Gamma }_{i,i^{\prime },j,j^{\prime }}a_i^{†}a_{i^{\prime }}\rho a_{j^{\prime }}^{†}a_j+\sum _i{\Lambda }_ia_i\rho a_i^{†},$

where [ ] is the commutator, ρ is the density matrix, H_BH is the Hamiltonian describing the dissipative-driven hardware, H_LS is the Lamb shift, H_decoh is an anti-Hermitian term proportional to the dechoerence rate of the hardware that leads to relaxation in the fixed excitation manifold and can be the Fourier transform of the bath correlation functions; Γ_{i,i′,j,j′}, is a tensor describing the quantum jump rate among fixed-excitation manifolds, and Λ_i is a tensor describing quantum jump rates between fixed-excitation manifolds.

The dechoerence of the hardware is gradually increased to drive the dynamics of the hardware to a classical regime steady state of dissipative dynamics that encodes the solution to the computational problem (step 404). After increasing the dechoerence, the dynamics of the dissipative-driven hardware can be simplified to:

$\frac{\rho }{t}=-2H_{\mathrm{decoh}}\rho +\sum _{i,i^{\prime },j,j^{\prime }}{\Gamma }_{i,i^{\prime },j,j^{\prime }}a_i^{†}a_{i^{\prime }}\rho a_{j^{\prime }}^{†}a_j$

Local marginal probabilities can then be determined by the hardware and in some implementations a classical computer (step 406):

$\frac{\mathrm{tr}\left[P_m\rho \right]}{t}=-2\mathrm{tr}\left[P_mH_{\mathrm{decoh}}\rho \right]+\sum _{i,i^{\prime },j,j^{\prime }}{\Gamma }_{i,i^{\prime },j,j^{\prime }},\mathrm{tr}\left[a_j^{†}a_{j^{\prime }}P_ma_{j^{\prime }}^{†}a_{i^{\prime }}\rho \right]$

where tr[ ] is the trace operation which in a density-matrix formulation is used to determine the expectation value of an operator, and P_m is a projector operator corresponding to the occupation density of a local cavity mode m.

The second term above retains density-density interactions between photons in a cavity mode i and in a cavity mode j that contribute to the number of photons in the visible cavity mode m. The second term further retains the Γ_{i,i′,j,j′} tensor which can be related to a Markov transition matrix, which is a matrix used in the problem if the problem can be described as a Markov Random Field.

In some implementations, the problem, e.g., a probabilistic inference, can be encoded in a quantum probability distribution of the dissipative Bose-Hubbard Hamiltonian or its extended engineered version; that is using the concept of quantum graphical models.

While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various hardware modules and components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and hardwares can generally be integrated together in a single software product or packaged into multiple software products.

Particular embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. An apparatus comprising:

a first plurality of superconducting cavities each configured to receive a plurality of photons;

a second plurality of superconducting cavities each configured to receive a plurality of photons; and

a plurality of couplers, wherein each coupler couples one superconducting cavity from the first plurality of superconducting cavities with one superconducting cavity from the second plurality of superconducting cavities such that the photons in the coupled superconducting cavities interact; and

wherein a first superconducting cavity of the first plurality of superconducting cavities is connected to a second superconducting cavity of the second plurality of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first plurality of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one or more of the other superconducting cavities of the second plurality of superconducting cavities to which the first superconducting cavities are coupled.

2. The apparatus of claim 1, wherein each coupler is configured to annihilate a photon in one superconducting cavity and create a photon in a different superconducting cavity.

3. The apparatus of claim 1, wherein at least one of the couplers comprises a Josephson Junction.

4. The apparatus of claim 1, wherein a Hamiltonian characterizing the apparatus is:

Σihini+Σi,jtij(αi†αj+h.c.)+ΣiUini(ni−1),

where ni is a particle number operator and denotes occupation number of a cavity mode i, αi† is a creation operator that creates a photon in cavity mode i, αj is an annihilation operator that annihilates a photon in cavity mode j, hi corresponds to a site disorder, Ui corresponds to an on-site interaction, ti,j are the hopping matrix elements, and h.c. is hermitian conjugate.

5. The apparatus of claim 4, wherein the plurality of couplers is trained to produce an output desired probability density function at a subsystem of interest at an equilibrium state of the apparatus.

6. The apparatus of claim 4, wherein the apparatus is trained as a Quantum Boltzmann Machine.

7. The apparatus of claim 1, wherein a Hamiltonian characterizing the apparatus is: ∑ i   h i  n i + ∑ i, j   t ij  ( a i †  a j + h. c. ) + ∑ i   U i  n i  ( n i - 1 ) + ∑ i, j   U ij  n i  n j,

wherein ni is a particle number operator and denotes occupation number of a cavity mode i, αi† is a creation operator that creates a photon in cavity mode i, αi is an annihilation operator that annihilates a photon in cavity mode j, hi corresponds to a site disorder, Ui corresponds to an on-site interaction, ti,j are the hopping matrix elements, and h.c. is hermitian conjugate.

8. The apparatus of claim 7, wherein the apparatus is operable to evolve adiabatically to a ground state of a problem Hamiltonian Hp=Σihini+ΣiUini(ni−1)+Σi,jUijninj.

9. The apparatus of claim 7, wherein the apparatus is operable to evolve adiabatically from a Mott-insulator state to a superfluid state, and wherein an initial Hamiltonian of the apparatus is Hi=Σi,jtij(αi†αj+h.c.).

10. The apparatus of claim 7, wherein the apparatus is operable to evolve adiabatically from a Mott-insulator state to a ground state of a problem Hamiltonian Hp=Σihini+ΣiUini(ni−1)+Σi,jUijninj, and wherein an initial Hamiltonian of the apparatus is Hi=Σi,jtij; (αi†αj+h.c.

11. The apparatus of claim 1, wherein the apparatus is configured to respond to an external field ε(t) and a Hamiltonian characterizing the apparatus in the external field is:

Σihini+Σi,jtij(αi†αj+h.c.)+ΣiUini(ni−1)+Σi[ε(t)αi†+ε(t)*αi]+HSB,

wherein

HSB=ΣiΣυ[κi,υ(αibυ†+αi†bυ)+λi,υαi†αi(bυ+bυ†)], and

wherein ni is a particle number operator, ε(t) is a slowly-varying envelope of an externally applied field to compensate for photon loss, HSB is a Hamiltonian of the interaction between the apparatus and a background bath in which the apparatus is located, bυ, and bυ† are annihilation and creation operators for a bosonic background bath environment, κi,υ is a strength of apparatus-bath interactions corresponding to exchange of energy, hi corresponds to a site disorder, Ui corresponds to an on-site interaction, ti,j are the hopping matrix elements, and λi,υ corresponds to a strength of local photon occupation fluctuations due to exchange of phase with the bath.

12. The apparatus of claim 11, wherein the apparatus is operable to be dissipatively-driven to a ground state of a problem Hamiltonian.

13. The apparatus of claim 1, wherein at least one cavity is a 2D cavity.

14. The apparatus of claim 1, wherein each cavity is a 3D cavity.

15. The apparatus of claim 1, wherein each superconducting cavity in the first plurality of superconducting cavities is connected to a superconducting cavity in the second plurality of superconducting cavities.

16. A method comprising:

providing the apparatus of claim 1 in an initial Mott-insulated state;

causing a quantum phase transition of the apparatus from the initial Mott-insulator state to a superfluid sate; and

adiabatically guiding the apparatus to a problem Hamiltonian.

17. The method of claim 16, further comprising:

causing a quantum phase transition of the apparatus from the superfluid state to a final Mott-insulator state; and

reading the state of each superconducting cavity in the apparatus.