Quantum generative adversarial network with provable convergence

Abstract

To provide a quantum generative adversarial network with provable convergence.SOLUTION: The present invention is related to a method and an apparatus for learning a target quantum state. In one aspect, a method for training a quantum generative adversarial network (QGAN) to learn a target quantum state includes a step iteratively adjusting parameters of the QGAN until a value of a QGAN loss function converges. Each iteration comprises the steps for: performing an entangling operation on a discriminator network input of a discriminator network in the QGAN to measure a fidelity of the discriminator network input, the discriminator network input including the target quantum state and a first quantum state output from a generator network in the QGAN, the first quantum state approximating the target quantum state; and performing a minimax optimization of the QGAN loss function to update the QGAN parameters, the QGAN loss function being dependent on the measured fidelity of the discriminator network input.SELECTED DRAWING: Figure 2

Description

Technical Field

[0001] This specification relates to quantum computing and generative adversarial networks.

Background Art

[0002] Classical computers have memory consisting of bits, where each bit can represent either 0 or 1. Quantum computers hold a sequence of quantum bits called qubits, where each qubit can represent 0, 1, or any quantum superposition of 0 and 1. Quantum computers operate by setting qubits to an initial state and controlling the qubits, for example, according to a sequence of quantum logic gates.

[0003] Generative adversarial networks are a form of generative machine learning that achieve state-of-the-art performance in various high-dimensional complex tasks, including the generation of realistic images like photos, super-resolution, and molecular synthesis. Assuming access only to a training dataset S = {x data} sampled from the underlying data distribution p i}, GANs can generate realistic examples outside of S. Some probability distributions are classically difficult to sample, and thus, learning an accurate representation of any distribution p data (x) can benefit from access to quantum computing resources.

Summary of the Invention

Means for Solving the Problems

[0004] This specification describes a quantum generative adversarial network with provable convergence.

[0005] Generally, one innovative aspect of the subject matter described in this specification can be implemented in a method for training a quantum adversarial generative network to learn a target quantum state. The method includes iteratively adjusting the parameters of the quantum adversarial generative network until the value of the quantum adversarial generative network loss function converges. Each iteration includes performing an entangling operation on the discriminator network input of the quantum adversarial generative network to measure the fidelity of the discriminator network input, where the discriminator network input includes the target quantum state and a first quantum state output from the generator network of the quantum adversarial generative network, and the first quantum state approximates the target quantum state. The method also includes performing a minimax optimization of the quantum adversarial generative network loss function to update the parameters of the quantum adversarial generative network, where the quantum adversarial generative network loss function depends on the measured fidelity of the discriminator network input.

[0006] Other implementations of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. One or more classical and / or quantum computer systems can be configured to perform particular operations or actions by having software, firmware, hardware, or combinations thereof installed on the system that cause the system to perform the actions during operation. One or more computer programs can be configured to perform particular operations or actions by including instructions that, when executed by a data processing apparatus, cause the apparatus to perform the actions.

[0007] The above and other implementations can optionally include one or more of the following features, either alone or in combination. In some implementations, the value of the quantum adversarial generation network loss function converges to a Nash equilibrium.

[0008] In some implementations, each iteration is a step of processing an initial quantum state by a generator network to output a first quantum state, where the processing step includes applying a first quantum circuit to the initial quantum state, and i) the first quantum circuit is a parameterized quantum circuit, and the first quantum circuit parameters constitute generator network parameters included in the parameters of the quantum adversarial generation network.

[0009] In some implementations, the first quantum circuit has a circuit depth smaller than that of the quantum circuit used to generate the target quantum state.

[0010] In some implementations, the entanglement operation includes a parameterized entanglement operation that approximates a swap test.

[0011] In some implementations, the entanglement operation includes an ancilla-free swap test.

[0012] In some implementations, the ancilla-free swap test approximates an exact swap test and includes a second quantum circuit, where the second quantum circuit is a parameterized quantum circuit, and the second quantum circuit parameters constitute discriminator network parameters included in the parameters of the quantum adversarial generation network.

[0013] In some implementations, the quantum adversarial generation network loss function includes one minus the measured fidelity of the discriminator network input.

[0014] In some implementations, the step of performing min-max optimization of the quantum adversarial generation network loss function includes fixing the generator network parameters to the values determined in the previous iteration and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameters to determine the updated values of the discriminator network parameters for the iteration, and fixing the discriminator network parameters to the updated values of the discriminator network parameters for the iteration and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the updated values of the generator network parameters for the iteration.

[0015] In some implementations, the method includes fixing the discriminator network parameters to values corresponding to a complete swap test and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the first updated values of the generator network parameters for the iteration, and fixing the generator network parameters to the first updated values and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameters to determine the updated values of the discriminator network parameters for the iteration, and fixing the discriminator network parameters to the updated values of the discriminator network parameters for the iteration and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the updated values of the generator network parameters for the iteration.

[0016] In some implementations, the target quantum state includes a superposition state, and the method further includes generating, by the generator network, a target quantum state to approximate a quantum random access memory according to the trained generator network parameters.

[0017] In some implementations, the method further includes training a quantum neural network using the generated target quantum state.

[0018] In some implementations, the step of iteratively adjusting the parameters of the quantum adversarial generative network until the value of the quantum adversarial generative network loss function converges generates the trained generator network parameters and discriminator network parameters, and the method further includes the step of using the generator network to generate a target state according to the trained generator network parameters.

[0019] In some implementations, the step of performing minimax optimization of the quantum adversarial generative network loss function to update the parameters of the quantum adversarial generative network includes the step of performing a plurality of circuit evaluations to calculate the gradients of the parameters of the quantum adversarial generative network.

[0020] Generally, another innovative aspect of the subject matter described herein is a quantum adversarial generative network system implemented by one or more quantum computers, wherein the quantum adversarial generative network is a discriminator network configured to perform an entanglement operation on the discriminator network input to measure the fidelity of the discriminator network input, the discriminator network input including a target quantum state and a first quantum state output from a generator network included in the quantum adversarial generative network system, the first quantum state approximating the target quantum state, and the quantum adversarial generative network system may be implemented including the discriminator network.

[0021] The subject matter described herein may be implemented in a particular manner to achieve one or more of the following advantages.

[0022] The currently described entangling quantum generative adversarial network (EQ-GAN) has been proven to converge to a global optimal Nash equilibrium and converges in problem instances where conventional quantum adversarial generative networks (QGANs) fail.

[0023] Furthermore, the task of learning a quantum circuit for generating an unknown quantum state can also be solved by a fully supervised approach. Instead of adversarially training a discriminator to distinguish between fake data and real data, the discriminator can be frozen to perform an accurate swap test and measure the fidelity of the state between the true and fake data. This replicates the original state in the absence of noise, but gate errors in the implementation of the discriminator cause convergence to the wrong optimal value. The adversarial approach of EQ-GAN is shown to be more robust to such errors than simpler supervised learning approaches. Training a quantum machine learning model can require a significant amount of time to compute gradients on current quantum hardware, so resilience to the drifting of gate errors during the training process is particularly valuable in the noisy intermediate-scale quantum (NISQ) era of quantum computing.

[0024] Furthermore, applications of EQ-GAN in the broader context of quantum machine learning for classical data are provided. Most quantum machine learning algorithms that promise exponential speedups over classical machine learning algorithms require quantum random access memory (QRAM). By learning a shallow quantum circuit for generating superpositions of classical data, EQ-GAN can be used to create an approximate QRAM. Such an application of QRAM for quantum neural networks can be shown to improve the classification accuracy of quantum neural networks with the same amount of training time over the accuracy of classification using classical datasets. Once trained, the quantum neural network can be easily inverted to provide interpretability of its classification process. EQ-GAN provides a new paradigm for loading classical data into a quantum state prepared by a shallow quantum circuit through variational circuit optimization.

[0025] Details of one or more implementations 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

[0026]

Figure 1

Figure 2

Figure 3

Figure 4A

Figure 4B

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

DETAILED DESCRIPTION OF THE INVENTION

[0027] Like reference numbers and designations in the various drawings indicate like elements.

[0028] An adversarial generative network (GAN) includes a parameterized generator network G(θ g , z) and a parameterized discriminator network D(θ d ; z). The generator converts a vector sampled from the input distribution z~p0(z) into a data example G(θ g , z), thus converting p0(z) into a new distribution p g (z) of fake data. The discriminator takes an input sample x and gives the probability D(θ d ; z) that the sample is real (from the data) or fake (from the generator network). Training corresponds to a minimax optimization problem, and alternately improves the discriminator's ability to distinguish real / fake samples and the generator's ability to fool the discriminator. For example, with respect to the cost function V given by the following equation (1),

[0029]

Equation

[0030] is solved.

[0031]

Equation

[0032] In Equation (1), θ grepresents the generator network parameters, θ d represents the discriminator network parameters, p data p(z) represents the true data distribution, and p0(z) represents the input distribution.

[0033] When G and D have sufficient capabilities, for example, when approaching the space of any function, the global optimum of this minimax game exists, and it is proven that p g (x) = p data (x) corresponds uniquely. Multilayer perceptrons can be used to parameterize D and G, but it is also possible to increase the dimension of the function space by replacing classical neural networks with quantum neural networks. In the most common case, classical data can be represented by the density matrix σ = Σ i p i |ψ i ><ψ i |, where p i ∈[0, 1] represents a real number within a determined positive range, and |ψ i > is an orthogonal basis state. In the first proposal of the Quantum GAN (QGAN), the generator network is defined by a quantum circuit U that outputs the quantum state ρ = U(θ g )ρ0U † (θ g ) from the initial state ρ0. The discriminator obtains either the true data σ or the fake data ρ, performs a positive operator valued measurement (POVM), and returns either the true data operator T or the fake data operator F,

[0034]

Number

[0035] is. Therefore, the probability that any state ρ in is true data is

[0036] [Mathematics]

[0037] is given by. QGAN solves, for example, the min-max game given by the following equation (3).

[0038] [Mathematics]

[0039] Since the set of positive operators with 1-norm less than or equal to 1 is convex and compact, gradient descent can be used to optimize the discriminator's measurement. The optimal discriminator measurement is given by the Helstrom measurement, and the operators P + (σ - ρ) and 1 - P + (σ - ρ) distinguish the positive and negative parts of σ - ρ. That is, the strictly positive eigenvalues

[0040] [Mathematics]

[0041] and the strictly negative eigenvalues

[0042] [Mathematics]

[0043] (corresponding eigenstates

[0044] [Mathematics]

[0045] and

[0046] [Mathematics]

[0047] (having

[0048]

number

[0049] Given, the optimal classifier is

[0050]

number

[0051] and F = 1 - P + (σ-ρ) is selected. D(θ d , σ) - D(θ d , ρ(θ g )) = Tr[Tσ] - Tr[Tρ(θ g )] > 0 to reach a Nash equilibrium, the generator g θ ) increases g must be modified. Some methods propose the update ρ→ρ' = ρ + α(σ-ρ) for α > 0 by minimizing equation (3), but this strategy does not yield a Nash equilibrium. The evaluation of the trace by the T operator aligns the generated data only to the positive projection of σ-ρ. This eventually leads to a mode collapse, as shown in the example below.

[0052] Consider a generator initialized with the true data states σ and ρ, and each state is

[0053]

number

[0054]

number

[0055] defined by, where σ x represents the Pauli operator x, and σ y represents the Pauli operator y. Maximizing Equation (3) in the Helstrom measurement by decomposing σ-ρ = σ y / 2, the discriminator obtains

[0056]

Number

[0057] When optimizing over the space of density matrices, the generator rotates ρ to be parallel to T and

[0058]

Number

[0059] also gives. In the next iteration, the discriminator attempts to perform a new Helstrom measurement to distinguish σ from ρ', which results in T' = P + (σ - ρ') = ρ. When the generator is readjusted to fit the new measurement operator, ρ'' = ρ. Now, if the QGAN is trained to completely solve the minimax optimization problem at each iteration, it is easy to see that it does not converge. Instead, it always oscillates between the states ρ' and ρ, and neither of those states is a Nash equilibrium of the minimax game for the performance of the QGAN under such mode collapse.

[0060] Figure 1 is a graph 100 showing the performance of a conventional QGAN that learns the state defined in Equation (5) with the initialization given by Equation (6). The x-axis represents the number of training episodes. The y-axis represents the loss. Graph 100 shows that mode collapse appears as oscillations in the losses of the generator and discriminator without converging to the global optimum.

[0061] More generally, consider oscillations between a finite set of states. The function T σ\(\rho = P\) + \((\sigma - \rho)\) is the optimal Hellstrom measurement obtained from the positive part of the spectral decomposition of \(\sigma - \rho\) in Equation (4).

[0062]

Number

[0063] is represented as follows.

[0064]

Number

[0065] If \(T\) is the \(k\)-fold composition of \(T\) with itself, and there exists some \(k>1\) such that \(T = \rho\), then it is sufficient to guarantee oscillations between \(k\) states. For an \(n\)-qubit system, this can be achieved by preparing target and initial states separated by an angle \(\pi / 3\) on the generalized Bloch sphere. (k) This problem only consistently exists when the discriminator of the QGAN is allowed to converge to the Hellstrom measurement during training, which may make the QGAN architecture more sensitive to the choice of hyperparameters, especially the learning rate and the number of epochs at which the discriminator and generator are trained in each iteration. When the discriminator and generator of the QGAN are fully trained, mode collapse is shared by both \(\rho\) and \(\rho'\), thus resulting in a constant fidelity of 3 / 4 throughout the oscillations. However, even in a regime with standard learning rate and only 1 epoch per iteration, the QGAN oscillates at the beginning of training. Unstable training is difficult to overcome even in classical GAN architectures, and thus progress in understanding ways to prevent such non-convergence is important for both quantum and classical machine learning.

[0066]

[0067] ​This specification describes a new quantum GAN that does not experience the above-mentioned mode collapse and thus provides a more robust QGAN architecture. The new quantum GAN is an entanglement QGAN (referred to herein as EQ-GAN) that entangles both real data and fake data instead of providing the discriminator with either real data or fake data.

[0068] Exemplary Operating Environment FIG. 2 is a block diagram of an entanglement quantum adversarial network (EQ-GAN) 200.

[0069] EQ-GAN 200 includes a real data state generator 202. The real data state generator 202 includes quantum hardware configured to generate a target quantum state, e.g., the state 208 of real data. In some implementations, the real data state generator 202 can prepare the target quantum state by applying a quantum circuit to an initial quantum state. In some cases, e.g., if the target quantum state is a superposition of classical data, the quantum circuit required to generate a particular target quantum state may include quantum logic gates that are costly to implement and / or may have a large circuit depth. Thus, generating a large number of target quantum states can be inefficient or infeasible.

[0070] EQ-GAN 200 also includes a generator network 204. The generator network 204 is configured to generate a quantum state that approximates a target quantum state, such as the state 210 of fake data. For example, as described in more detail below, the generator network may include quantum computing hardware configured to apply a parameterized quantum circuit to an initial quantum state in order to output a quantum state that approximates the target quantum state. The parameterized quantum circuit may have a smaller circuit depth compared to the quantum circuit implemented by the true data state generator 202 for generating the exact target quantum state. Thus, training the generator network 204 by adjusting the parameterized quantum circuit parameters until the value of the loss function of the EQ-GAN converges can enable the generator network 204 to generate an accurate approximation of the target quantum state at a lower computational cost. Exemplary operations performed by the generator network 204 are described in more detail below with reference to FIGS. 3-11. Exemplary hardware included in the generator network 204 is described in more detail below with reference to FIG. 12.

[0071] EQ-GAN 200 also includes a discriminator network 206. The discriminator network 206 is configured to receive a discriminator network input and perform a entanglement operation 214 on the discriminator network input to measure the fidelity 212 of the discriminator network input. The discriminator network input includes a true data state 208 obtained from a true data state generator 202 and a false data state 210 output from a generator network 204. That is, the discriminator network 206 is configured to entangle the true data state and the false data state. The entanglement operation is a parameterized entanglement operation that approximates a swap test. In some implementations, the entanglement operation requires an ancilla qubit. In other implementations, the entanglement operation is an approximation of the swap test without an ancilla. In either case, the entanglement operation can be implemented by applying a parameterized quantum circuit. Exemplary operations performed by the discriminator network 206 are described in more detail below with reference to FIGS. 3-11. Exemplary hardware included in the discriminator network 206 is described in more detail below with reference to FIG. 12.

[0072] EQ-GAN 200 can be trained to enable the generator network 204 to learn a quantum circuit that generates an improved approximation of a target quantum state. During training, the generation of the false data state by the generator network 204 and the learning of the fidelity measurement by the discriminator network 206 are adversarially optimized until a convergence criterion is met. Once trained, the generator network 204 can be used to generate an approximation 216 of the target quantum state according to the trained generator network parameters, for example, to approximate a quantum random access memory. An exemplary process for training EQ-GAN to learn a target quantum state is described below with reference to FIG. 3.

[0073] An exemplary process for training EQ-GAN FIG. 3 is a flowchart of an exemplary process 300 for training a quantum adversarial generation network to learn a target quantum state, where the quantum adversarial generation network includes a generator network and a discriminator network. For convenience, process 300 is described as being executed by quantum hardware that communicates with control electronics located at one or more locations. For example, system 200 of FIG. 2 appropriately programmed according to this specification may execute process 300.

[0074] The system iteratively adjusts the parameters θ g 、θ d of the quantum adversarial generation network until the value of the quantum adversarial generation network loss function converges. The quantum adversarial generation network loss function is described below with reference to Equation (8).

[0075] In each iteration, the generator network processes an initial quantum state ρ0 to output a quantum state ρ (step 302). The processing can include applying a first quantum circuit U to the initial quantum state, where the first quantum circuit is a parameterized quantum circuit and includes parameters that constitute the generator network parameters θ g . That is, the quantum state can be given by ρ = U(θ g )ρ0U † (θ g ), where U(θ g ) represents the parameterized first quantum circuit, θ g represents the generator network parameters, and ρ0 represents the initial quantum state. The quantum state ρ approximates the target quantum state, and iteratively adjusting the generator network parameters θ g enables the first quantum circuit U(θ g ) to generate a better approximation to the target quantum state. The processing may be executed using quantum hardware.

[0076] In each iteration, the discriminator network performs a entanglement operation on the discriminator network input to measure the fidelity of the discriminator network input (step 304). The discriminator network input includes the target quantum state and the quantum state output from the generator network. That is, the discriminator network is not directly similar to the discriminator of a classical GAN. Instead of individually evaluating whether the data is fake or real, the discriminator is always provided access to the true data σ and, as given by the following equation (7), performs a fidelity measurement on the input state ρ in for the

[0077]

Number

[0078] .

[0079]

Number

[0080] This enables the loss function of quantum adversarial generation to converge to a Nash equilibrium. The entanglement operation may be performed using quantum hardware.

[0081] The entanglement operation performed by the discriminator network is a parameterized entanglement operation that approximates the swap test. The swap test is a quantum computing procedure used to check how different two quantum states are. The swap test requires, for example, an auxiliary qubit initialized in the 0 state, and repeatedly applies a Hadamard gate to the auxiliary qubit, applies a CSWAP (also called a controlled-swap gate or Fredkin gate) gate to pairs of qubits from the first and second quantum states, applies a Hadamard gate to the auxiliary qubit, and measures the auxiliary qubit, for example, in the Z basis to determine how different the quantum states are.

[0082] In some implementations, the discriminator network D σ (θ d , ρ in ) can be a parameterized quantum circuit that uses an auxiliary qubit (similar to the case of the exact swap test), and the circuit parameters constitute the discriminator network parameters. The parameters that implement the exact swap test

[0083] [Number]

[0084] exist, that is,

[0085] [Number]

[0086] when it is, D σ has sufficient expressiveness to reach an optimal discriminator during optimization. However, the conventional swap test across two n-qubit states requires two-qubit gates spanning 2n qubits, so implementation on a quantum device with local connectivity incurs an exorbitant overhead in circuit depth. Thus, in some implementations, the discriminator network can be a parameterized circuit that uses an auxiliary qubit to approximate the swap test.

[0087] FIG. 4A is a circuit diagram 400 showing an exemplary discriminator network architecture. The exemplary discriminator network architecture includes three quantum states 402a - c. The first quantum state 402a is an auxiliary qubit prepared in an initial state, e.g., the 0 state. The second quantum state 402b is the output ρ(θ g ) of the generator network, e.g., fake data. The third quantum state 402c is the target quantum state σ of one or more qubits, e.g., real data.

[0088] The discriminator network applies a first Hadamard gate 404 to the auxiliary qubit 402a and a unitary operator 406 to the auxiliary qubit 402a, the output ρ(θ g ) of the generator network, and the target quantum state σ. The unitary operator 406 depends on a set of discriminator parameters θ d and approximates a swap test. The unitary operator 406 can represent a sequence of quantum logic gates, and the quantum logic gates included in the sequence can vary based on the sizes of the second and third quantum states and a specific hardware implementation. For example, if the target quantum state is a single-qubit state and the output of the generator network is a single-qubit state, the sequence of quantum logic gates can potentially include single-qubit rotation gates, S gates, T gates, Hadamard gates, Pauli X gates, and CZ gates. An exemplary circuit representation of the unitary operator 406 is shown in FIG. 4B. In FIG. 4B, X1 to X7 represent free parameters to be trained.

[0089] The discriminator network further applies a second Hadamard gate 408 to the auxiliary qubit 402a and measures the auxiliary qubit using a measurement operation 410 to obtain a discriminator output 412 representing the difference between the second quantum state 402b and the third quantum state 402c.

[0090] To further simplify the physical implementation of the discriminator network, in some implementations, the discriminator network D σ (θ d , ρ in ) can be a parameterized circuit that does not include an auxiliary qubit. Instead, the discriminator network can be a parameterized circuit that performs a destructive (without auxiliary, without auxiliary qubit) approximation to the swap test.

[0091] For example, for a quantum device with planar connectivity, the CNOT gate can be decomposed into operations. The CZ gate has unstable errors that can be effectively modeled using unknown-angle Z rotations on either qubit. The EQ-GAN formalism currently being described can overcome single-qubit phase errors by applying an RZ(θ) gate immediately after each CZ operation. During adversarial training, the free angle θ is optimized by gradient descent to reduce the errors of the two-qubit gate. Thanks to the convergence properties provided by the adversarial generation framework, the discriminator converges provably to the discriminator of the best possible state. This serves as the motivation for early stopping (as shown in Figure 7) when the loss of the discriminator indicates that the discriminator of the best state has been reached.

[0092]

Number

[0093]

[0094] ​FIG. 5 is a circuit diagram 500 of an approximate swap test without an auxiliary qubit between a first three - qubit state 502 and a second three - qubit state 504. The left side of the circuit diagram 500 shows a precise swap test 506. In the precise swap test, a first Hadamard gate 508 is applied to an auxiliary qubit 510, and three CSWAP gates 512 are applied to pairs of qubits of the first and second three - qubit states 502 and 504. For example, the first CSWAP gate is applied to the first qubit of the first state 502 and the first qubit of the second state 504, the second CSWAP gate is applied to the second qubit of the first state 502 and the second qubit of the second state 504, the third CSWAP gate is applied to the third qubit of the first state 502 and the third qubit of the second state 504, and the auxiliary qubit 510 acts as a control for each CSWAP gate. A second Hadamard gate 514 is applied to the auxiliary qubit 510, and a measurement operation 516 is performed to obtain the measured result of the auxiliary qubit.

[0095] The right side of the circuit diagram 500 shows an alternative implementation of the swap test 518. By rewriting the controlled - swap operation 512 as CNOT gates 520 applied to each qubit of the first state 502, where each qubit of the second state 504 acts as a control for each CNOT gate, Hadamard gates 522 applied to each qubit of the second state 504, measurement operations 524 performed on each qubit of the first state 502 and the second state 504, and Toffoli gates 526 applied to an ancilla classical bit, where each Toffoli gate uses each qubit of the first state 502 and each qubit of the second state 504 as controls, and replacing the computational - basis operations with classical post - processing, the swap test can be performed without an auxiliary qubit using an ancillary classical bit (hence the term "auxiliary - free" swap test).

[0096] Returning to FIG. 3, the system performs a minimax optimization of the quantum adversarial generation network loss function to update the parameters of the quantum adversarial generation network (step 306). The minimax optimization may be performed using one or more classical processors. The quantum adversarial generation network loss function depends on the discriminator network output (the measured fidelity D σ (θ d , ρ(θ g ))) and is equal to one minus the measured fidelity of the discriminator network input, as given by the following equation (8).

[0097]

Equation

[0098] Performing a minimax optimization of the quantum adversarial generation network loss function involves fixing the generator network parameter θ g to the value determined in the previous iteration (or the initial value if the iteration is the first iteration) and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameter θ d to determine the updated value of the discriminator network parameter for the iteration, and fixing the discriminator network parameter to the updated value of the discriminator network parameter θ d for the iteration and minimizing the quantum adversarial generation network loss function with respect to the generator network parameter θ g to determine the updated value of the generator network parameter for the iteration. That is, the EQGAN architecture adversarially optimizes the generation of the state ρ(θ g ) and the learning of the fidelity measurement D σ .

[0099] As described above in connection with steps 302 - 306, repeatedly adjusting the parameters of the quantum adversarial generation network until the value of the quantum adversarial generation network loss function converges generates the trained generator network and discriminator network parameters that define the trained generator network and discriminator network. Once trained, the generator network can generate an accurate approximation to the target state according to the trained generator network parameters, for example, as part of the QRAM described below.

[0100] Now, it is shown that there exists a unique Nash equilibrium at the desired location. By definition, 0 ≤ D σ (θ d , ρ(θ g )) ≤ 1 is the probability of measuring the state |1> at the end of the circuit shown in Figure 4A or Figure 5. When the discriminator implements the identity transformation, i.e.,

[0101]

Number

[0102] is the case, the probability of observing the state |1> is 0. In the first step of maximizing the discriminator, the discriminator performs a non-trivial entanglement operation on the generator output and the true data. Further, when the circuit ansatz for the swap test regarding U(θ d ) is given, the maximum value for distinguishing two arbitrary states is uniquely achieved by the angle of the complete swap test. Although the discriminator may not choose the swap test, the next step is to minimize D σ (θ d , ρ(θ g )) from the generator side. If the discriminator does not perform the swap test, the generator can choose a new arbitrary state that is not well-distinguished by the discriminator because it is not using the fidelity comparison. Finally, the generator can only improve if and when the discriminator uses the swap test, at which point a unique minimum value for ρin = σ.

[0103] The circuit parameterization U(θ d ) can be selected based on various factors including the type of device used to implement the discriminator network, the available connectivity within the device, the type of gates that can be efficiently implemented by the device, etc. For example, for near-term quantum devices with planar connectivity, fixed gates or two-qubit entangling gates can be efficiently implemented and thus can form the circuit parameterization.

[0104] A circuit parameterization that is not well-chosen results in a landscape of non-convex loss functions and may thus be difficult to optimize by gradient descent, which is a problem shared with QGANs because it is difficult to represent any unitary as a shallow quantum circuit. Similarly, non-convexity in classical GANs often hinders convergence. However, the EQ-GAN architecture converges successfully in intractable problem cases by a fully trained and appropriately parameterized QGAN. Figure 6 is a graph 600 showing a comparison between a QGAN that learns the quantum state given by Equation (5) and the currently described EQ-GAN. The x-axis represents the number of iterations. The y-axis represents the overlap with the state of the data. Graph 600 shows that while the QGAN oscillates infinitely between two states of equal fidelity (3 / 4), the EQ-GAN converges rapidly to full fidelity.

[0105] Learning error suppression EQ-GAN can achieve improved robustness against gate errors compared to simpler supervised learning approaches for learning unknown quantum states. Instead of adversarially training the parameterized swap test used as a discriminator in EQ-GAN, the full swap test can be applied iteratively by a frozen discriminator. This may also cause the generator circuit to converge to the true data since the swap test guarantees a unique global optimum.

[0106] However, if there are gate errors in the swap test, this unique global optimum is offset from the true data. Since EQ-GAN does not rely on the exact parameterization of the full swap test, appropriate ansätze can learn to correct for coherent errors observed in near-future quantum hardware. In particular, gate parameters such as the conditional Z phase, single qubit Z phase, and swap angle in two-qubit entanglement gates can drift and oscillate on the order of O(10) timescales. Such unknown systematic and time-dependent coherent errors pose significant challenges for applications in quantum machine learning where gradient calculation and update require many measurements.

[0107] Large deviations in single-qubit and two-qubit Z rotation angles can be significantly reduced by including additional single-qubit Z phase compensation. The effectiveness and importance of mitigating such systematic errors were demonstrated recently when achieving state-of-the-art accuracy in the energy estimation of fermionic molecules. In learning the discriminator circuit closest to the true swap test, the adversarial learning of EQ-GAN provides a useful paradigm that may be widely applicable to improving the fidelity of other near-future quantum algorithms.

[0108] Let the unitary of the adversary discriminator be given by U(θ d ), where

[0109]

Number

[0110] corresponds to the perfect swap test in the absence of noise. Given a noisy channel ε that is trace-preserving completely positive, the discriminator is replaced by a new unitary operation

[0111]

Number

[0112] . The supervised method applies an approximate swap test given by

[0113]

Number

[0114] , while the adversarial swap test generally exhibits better performance when there exist parameters

[0115]

Number

[0116] such that

[0117]

Number

[0118] . Since the discriminator defines a loss landscape optimized by the generator, the noisy unitary

[0119]

Number

[0120] If the parameterization is general enough to sufficiently reduce errors, ρ(θ g ) generated by EQ-GAN may converge to a state closer to σ than is possible with supervised methods.

[0121] Since the discriminator must converge to the swap test at the optimal Nash equilibrium, convergence may be heuristically improved by two-phase training in the presence of noise. In the first phase, the unitary

[0122] [Number]

[0123] may be an incomplete swap test, but the discriminator is frozen with the parameters of the complete swap test and the generator is trained until the loss converges. In the second phase of training, the discriminator is allowed to change adversarially against the generator and the parameters

[0124] [Number]

[0125] are sought. In the context of gate errors, this second phase may result in a unitary closer to the true swap test.

[0126] In other words, performing the minimax optimization of the quantum adversarial generation network loss function involves fixing the discriminator network parameters to values corresponding to a complete swap test and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the first updated values of the iterative generator network parameters, fixing the generator network parameters to the first updated values and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameters to determine the updated values of the iterative discriminator network parameters, and fixing the discriminator network parameters to the updated values of the iterative discriminator network parameters and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the updated values of the iterative generator network parameters.

[0127] As an example, consider the task of learning superposition states

[0128]

Number

[0129] in a noisy quantum device. According to the above discovery method, EQ-GAN is trained using a frozen discriminator during the first half of training and adversarially trained during the second half. The discriminator is defined by a swap test using the CZ gate that provides the necessary two-qubit operation. However, to learn to correct gate errors, the discriminator adversarially learns the angle of a single-qubit Z rotation inserted immediately after the CZ gate. Therefore, EQ-GAN achieves a state overlap that is significantly better than that of a complete swap test state.

[0130] Table I shows the average errors after multiple runs of EQ-GAN and the supervised learner in the experimental device.

[0131]

Table 1

[0132] In Table I (Table 1), the comparison of EQ-GAN and supervised learning machines on a quantum device with 50 qubits, CZ gates, and any single-qubit gates shows that the error of EQ-GAN (i.e., 1 - the fidelity of the state to the true data) is significantly smaller than that of the supervised learning machine, indicating successful adversarial training of the swap test with suppressed error. The uncertainty indicates two standard deviations.

[0133] FIG. 7 shows a first graph 700 plotting the comparison of EQ-GAN and supervised learning machines implemented on a simulated quantum device, and a second graph 750 plotting the comparison of EQ-GAN and supervised learning machines implemented on a physical quantum device. In both graphs, the x-axis represents the number of iterations, and the y-axis represents the fidelity of the state to the true data. In the simulation, noise following a normal distribution regarding single-qubit rotations is applied with a systematic bias away from 0, forcing the discriminator of the supervised learning machine to converge to the wrong state. It has been experimentally confirmed that EQ-GAN converges to a higher state overlap by learning to correct such errors with additional single-qubit rotations. The converged EQ-GAN (dashed line) is determined by the iteration at which the loss of the discriminator reaches an extreme value.

[0134] Application to QRAM Many applications of quantum machine learning require a quantum random access memory (QRAM) to load data in superposition states. However, loading arbitrary states may require noisy controlled rotations, and preparing a superposition of an arbitrary set of n states may achieve an operation that is at best O(n). Given a suitable ansatz, EQ-GAN can be used to learn states that are approximately equivalent to superpositions of data. That is, the target quantum state described above with reference to FIG. 3 can be a superposition state representing a superposition of data, and when trained, the generator network can be used to generate a superposition state that approximates a QRAM according to the trained generator network parameters. Quantum acceleration can be obtained if the training of EQ-GAN has a lower computational cost than the number of calls required for a QRAM in the context of another algorithm.

[0135] To demonstrate the variational QRAM, a two-peak dataset sampled from different Gaussian distributions is used. Encoding the empirical probability density function exactly requires a very deep circuit and multiple-control rotations, but an ansatz of a shallow circuit that generates exponential peaks can be selected. FIG. 8 shows two variational QRAM ansätze for generating peaks. Class 0 corresponds to the central peak and Class 1 corresponds to the offset peak. When trained to approximate the empirical data distribution, the variational QRAM reproduces the original dataset fairly faithfully. FIG. 9 shows the full dataset with two peaks (sampled from a normal distribution, N = 120) and the variational QRAM of the training dataset (N = 60). The variational QRAM is obtained by training EQ-GAN to generate a state ρ with a shallow peak ansatz to approximate an exact superposition of the state σ. The training and test datasets (N = 60 each) are both balanced between the two classes.

[0136] As a proof of principle for using such a QRAM in the context of quantum machine learning, a quantum neural network can be trained using the QRAM described above, and the hinge loss is calculated either by considering each data entry (encoded as a quantum circuit) individually or by considering each class (encoded as a superposition in a variational QRAM) individually. Assuming the same number of circuit evaluations for calculating the gradients, as shown in Table II below, the superposition converges to a higher accuracy at the end of training, despite using an approximate distribution.

[0137] [Table 2]

[0138] Table II shows the test accuracy (N = 60) of a quantum neural network (QNN) trained on all samples of a training dataset (N = 60) for one epoch or trained in a variational QRAM for an equal number of circuit evaluations. The QNN trained in the variational QRAM did not have direct access to the original dataset, but the accuracy is evaluated on the raw dataset. Uncertainty indicates two standard deviations.

[0139] Differences obtained from experiments on the performance between training of a quantum neural network (QNN) by individual examples of classical datasets and training of the QNN by superposition of data obtained from a pre-trained EQ-GAN can be shown. Any parameterized circuit with single and two-qubit gates can be used to construct the ansatz of the QNN. Due to the planar connectivity of the quantum device with 50 qubits, CZ gates, and any single-qubit gates, the QNN shown in FIG. 10 can be implemented in the state of 4-qubit data. FIG. 10 shows an exemplary quantum neural network architecture (left) and its corresponding layout on the quantum device (right). The state of 4-qubit data is constructed with the circuit shown in FIG. 7 and placed in the |data> state on the blue qubits. Then, the readout qubit (orange) performs the parameterized two-qubit interaction shown in FIG. 11. In order to use the native CZ two-qubit gate,

[0140] [Number]

[0141] the rank-4 entanglement gate G given by is implemented, and this rank-4 entanglement gate G can be decomposed as shown in FIG. 11. Similar to some conventional proposals, instead of using the ZZ interaction, any two-qubit entanglement interaction can be freely selected to construct the parameterized unitary.

[0142] FIG. 11 shows the decomposition of the two-qubit entanglement gate G(θ) used in the ansatz of the QNN given by Equation (9).

[0143] The QNN can be trained in two ways - by sampling or by superposition. As described above, the superposition methodology should not use an exact superposition of the training dataset. Instead, the superposition methodology can use a shallow approximation obtained by pre-training the EQ-GAN. For a fair comparison, an equal number of queries to the quantum device are allowed. As a result, for N = 60 examples with 30 examples per class, training by sampling is performed for 1 epoch with 60, corresponding to 60 iterations executed on the quantum device. However, training by superposition evaluates the superposition of each class 30 times (since there are 2 classes) and also accesses the quantum device for 60 iterations. Additionally, Bayesian optimization is used to tune the different learning rates of the sampling and superposition methodologies. In the simulation, the Adam learning rate from 10 -4 to 10 -1 is optimized with 10 random parameter trials and 40 evaluations of the Gaussian process estimator. For each parameter query, the output of the QNN is averaged over 10 trials to reduce any statistical fluctuations. Then, the QNN using the final learning rates (10 -3.93 for sampling and 10 -1.83 for superposition) is evaluated over 50 trials to obtain the final performance reported in Table II (Table 2) along with the calculated standard deviation.

[0144] FIG. 12 shows an exemplary system 1200 for performing classical and quantum computing as described herein. The exemplary system 1200 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices at one or more locations where the systems, components, and techniques described herein may be implemented.

[0145] Exemplary system 1200 includes an exemplary quantum computing device 1202. The quantum computing device 1202 can be used to perform the quantum computing operations described herein according to some implementations. The quantum computing device 1202 is intended to represent various forms of quantum computing devices. The components shown herein, their connections and relationships, and their functions are merely exemplary and do not limit the implementations of the invention described and / or claimed herein.

[0146] Exemplary quantum computing device 1202 includes a qubit assembly 1252 and a control and measurement system 1204. The qubit assembly includes a plurality of qubits, such as qubit 1206, used to perform the operations of an algorithm or quantum computing. Although the qubits shown in FIG. 12 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. Qubit assembly 1252 also includes adjustable coupling elements, such as coupler 1208, that enable interaction between the coupled qubits. In the schematic depiction of FIG. 12, each qubit is adjustably coupled by its respective coupling element to each of its four adjacent qubits. However, this is an exemplary arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that enable coupling between non-adjacent qubits, and arrangements that include adjustable coupling between three or more qubits.

[0147] Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The particular physical realization of multiple qubits and how they interact with each other depends on various factors including the type of quantum computing device included in the exemplary system 1200, or the type of quantum computation the quantum computing device is performing. For example, in an atomic quantum computer, a qubit may be realized by an atom, molecule, or solid-state quantum system, such as a hyperfine atomic state. As another example, in a superconducting quantum computer, a qubit may be realized by a superconducting qubit or a semiconductor qubit, such as a superconducting transmon state. As another example, in an NMR quantum computer, a qubit may be realized by a nuclear spin state.

[0148] In some implementations, quantum computing can proceed by initializing qubits in a selected initial state and applying a sequence of unitary operators to the qubits. Applying a unitary operator to a quantum state can include applying a corresponding sequence of quantum logic gates to the qubits. Exemplary quantum logic gates include one-qubit gates such as Pauli X, Pauli Y, Pauli Z (also called X, Y, Z), Hadamard gate, S gate, rotation, two-qubit gates such as controlled X, controlled Y, controlled Z (also called CX, CY, CZ), controlled NOT gate (also called CNOT), controlled swap gate (also called CSWAP), and gates including three or more qubits such as the Toffoli gate. Quantum logic gates can be implemented by applying control signals 1210 generated by the control and measurement system 1204 to the qubits and couplers.

[0149] For example, in some implementations, the qubits of the qubit assembly 1252 may be frequency adjustable. In these examples, each qubit may have an associated operating frequency that can be adjusted by applying a voltage pulse via one or more drive lines coupled to the qubit. Exemplary operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubits can perform. For example, setting the operating frequency to the corresponding idling frequency may put the qubit in a state where it does not strongly interact with other qubits and may be used to perform single qubit gates. As another example, when qubits interact via a coupler with fixed couplings, the qubits can be configured to interact with each other by setting their respective operating frequencies to frequencies that depend on some gate that detunes them from their common interaction frequency. In other cases, for example, when qubits interact via an adjustable coupler, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between the qubits and by setting their respective operating frequencies to frequencies that depend on some gate that detunes them from their common interaction frequency. Such interactions may be performed to execute multiqubit gates.

[0150] The type of control signal 1210 used depends on the physical implementation of the qubit. For example, the control signal may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system.

[0151] Quantum computing can be accomplished by measuring the state of qubits using quantum observables such as X or Z, respectively, using each control signal 1210. The measurement causes a readout signal 1212 representing the measurement result to be transmitted back to the measurement and control system 1204. The readout signal 1212 may include RF, microwave, or optical signals, depending on the physical manner of the quantum computing device and / or qubits. For convenience, the control signal 1210 and the readout signal 1212 shown in FIG. 12 are shown to address only selected elements (i.e., the top and bottom rows) of the qubit assembly, but during operation, the control signal 1210 and the readout signal 1212 can address each element within the qubit assembly 1252.

[0152] The control and measurement system 1204 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 1252 described above and other classical subroutines or calculations. The control and measurement system 1204 includes one or more classical processors, such as classical processors 1214, connected by one or more data buses, one or more memories, such as memory 1216, and one or more I / O units, such as I / O unit 1218. The control and measurement system 1204 can be programmed to send a sequence of control signals 1210 to the qubit assembly, for example, to perform a selected series of quantum gate operations, and to receive a sequence of readout signals 1212 from the qubit assembly, for example, as part of performing a measurement operation.

[0153] The processor 1214 is configured to process instructions for execution within the control and measurement system 1204. In some implementations, the processor 1214 is a single-threaded processor. In other implementations, the processor 1214 is a multi-threaded processor. The processor 1214 can process instructions stored in the memory 1216.

[0154] Memory 1216 stores information within control and measurement system 1204. In some implementations, memory 1216 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. Optionally, memory 1216 may include a storage device, such as a hard disk device, an optical disk device, a storage device shared via a network by multiple computing devices (e.g., a cloud storage device), and / or any other such large-capacity storage device, that can provide large-capacity storage for system 1204.

[0155] Input / output device 1218 provides input / output operations to control and measurement system 1204. Input / output device 1218 can include a D / A converter, an A / D converter, and an RF / microwave / optical signal generator, transmitter, and receiver, thereby transmitting control signal 1210 to qubit assembly and receiving readout signal 1212 from qubit assembly, as appropriate for the physical approach of the quantum computer. In some implementations, input / output device 1218 can also include one or more network interface devices, such as an Ethernet card, serial communication devices, such as an RS-232 port, and / or wireless interface devices, such as an 802.11 card. In some implementations, input / output device 1218 can include a driver device configured to receive input data and transmit output data to other external devices, such as a keyboard, printer, and display device.

[0156] Although an exemplary control and measurement system 1204 is shown in FIG. 12, implementations of the subject matter and functional operations described in this specification may be implemented in other kinds of digital electronic circuitry, or in computer software, firmware, or hardware, or in combinations of one or more of them, including the structures disclosed herein and their structural equivalents.

[0157] Exemplary system 1200 includes an exemplary classical processor 1250. Classical processor 1250 may be used to perform classical computing operations described herein, such as classical machine learning methods described herein, in some implementations.

[0158] Implementations and operations of the subject matter described in this specification may be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry, or more generally quantum computing systems, tangibly embodied software or firmware, computer hardware, or combinations of one or more of them, including the structures disclosed herein and their structural equivalents. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0159] Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or additionally, the program instructions can be generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing apparatus, and can be encoded on an artificially generated propagated signal that encodes digital and / or quantum information, e.g., an electrical, optical, or electromagnetic signal generated by a machine.

[0160] The terms quantum information and quantum data refer to information or data carried by, held in, or stored in a quantum system, and the smallest significant system is the qubit, i.e., the system that defines the unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems can include, for example, multi-level systems with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis state is specified by the ground state and the first excited state, but it is understood that other setups are possible where the computational state is specified by a higher-level excited state.

[0161] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and includes, by way of example, all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, including programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The apparatus may be or further include dedicated logic circuits, such as FPGAs (field programmable gate arrays), ASICs (application specific integrated circuits), or quantum simulators, i.e., quantum data processing apparatuses designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a dedicated quantum computer that does not have the ability to perform universal quantum computing. Optionally, in addition to the hardware, the apparatus may include code for creating an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stacks, database management systems, operating systems, or combinations of one or more of them.

[0162] A digital computer program, which may also be referred to or termed as a program, software, software application, module, software module, script, or code, can be described in any form of programming language, including compiler-based languages or interpreter-based languages, or declarative languages or procedural languages, and can be deployed in any form, either as a stand-alone program or in the form of a module, component, subroutine, or other unit suitable for use within a digital computing environment. A quantum computer program, which may also be referred to or termed as a program, software, software application, module, software module, script, or code, is described in any form of programming language, including compiler-based languages or interpreter-based languages, or declarative languages or procedural languages, and can be converted into a suitable quantum programming language or can be described in a quantum programming language, such as QCL or Quipper.

[0163] A computer program may, but need not necessarily, correspond to a file in a file system. The program may be stored as part of a file that holds other programs or data, such as one or more scripts stored in a markup language document, a single file dedicated to the program in question, or multiple organized files, such as files that hold one or more modules, subprograms, or portions of code. A computer program can be stored on one computer, or placed in one location, or be deployed to be executed on multiple computers that are distributed across multiple locations and interconnected by digital and / or quantum data communication networks. A quantum data communication network is understood to be a network that may use a quantum system, such as qubits, to transmit quantum data. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network may transmit both quantum and digital data.

[0164] The processes and logical flows described herein can be performed by one or more programmable computers operating using one or more processors to perform operations on input data and generate output, and executing, as appropriate, one or more programs. The processes and logical flows can also be performed by, or by a combination of, dedicated logic circuitry, such as an FPGA or ASIC, or a quantum simulator, or by a combination of dedicated logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers, and the apparatus can be implemented as dedicated logic circuitry, such as an FPGA or ASIC, or a quantum simulator, or by a combination of dedicated logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers.

[0165] "Configured to" for a system of one or more computers to perform a particular operation or action means that the system has installed on it software, firmware, hardware, or a combination thereof that causes the system to perform the operation or action during operation. "Configured to" for one or more computer programs to perform a particular operation or action means that the one or more programs include instructions that cause an operation or action to be performed on a device when executed by a data processing device. For example, a quantum computer may receive instructions from a digital computer that cause an operation or action to be performed on the device when executed by a quantum computing device.

[0166] A computer suitable for the execution of a computer program can be based on a general purpose or special purpose processor, or any other kind of central processing unit. Generally, the central processing unit unit receives instructions and data from a read-only memory, a random access memory, or a quantum data, for example, a quantum system suitable for transmitting photons, or a combination thereof.

[0167] The elements of a computer include a central processing unit for executing or performing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be complemented by or incorporated into a dedicated logic circuit or a quantum simulator. Generally, a computer also includes one or more mass storage devices for storing data, such as magnetic disks, magneto-optical disks, optical disks, or a quantum system suitable for storing quantum information, or is operatively coupled to receive data from, transfer data to, or both of those mass storage devices. However, a computer may not have such devices.

[0168] A quantum circuit element (also referred to as a quantum computing circuit element) includes circuit elements for performing quantum processing operations. That is, a quantum circuit element is configured to perform operations on data in a non-deterministic manner by utilizing quantum mechanical phenomena such as superposition and entanglement. Certain quantum circuit elements, such as qubits, can be configured to represent information of two or more states simultaneously and operate on such information. Examples of superconducting quantum circuit elements include, among others, quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs: superconducting quantum interference device) (e.g., RF-SQUIDs or DC-SQUIDs).

[0169] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively execute the instructions of a computer program by performing basic arithmetic, logical, and / or input / output operations on data, and the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to send data to and / or receive data from quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuits, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and an energy-efficient version of RSFQ that does not use bias resistors, the ERSFQ device.

[0170] In certain cases, some or all of the quantum circuit elements and / or classical circuit elements may be implemented using, for example, superconducting quantum circuit elements and / or classical circuit elements. The fabrication of superconducting circuit elements may involve the deposition of one or more materials such as superconductors, dielectrics, and / or metals. Depending on the materials selected, these materials may be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among others, during the deposition process. The process for manufacturing the circuit elements described herein may involve removing one or more materials from the device during manufacturing. Depending on the materials removed, the removal process may include, for example, wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein may be patterned using known lithography techniques (e.g., photolithography or electron beam lithography).

[0171] During operation of a quantum computing system using superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled within a cryostat to a temperature that allows the superconducting material to exhibit superconducting properties. A superconductor (or superconducting) material can be understood as a material that exhibits superconducting properties below the superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature 1.2 Kelvin) and niobium (superconducting critical temperature 9.3 Kelvin). Thus, superconducting structures such as superconducting traces and superconducting ground planes are formed from materials that exhibit superconducting properties below the superconducting critical temperature.

[0172] In certain implementations, control signals for quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0173] Suitable computer-readable media for storing computer program instructions and data include, by way of example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices, magnetic disks such as internal hard disks or removable disks, magneto-optical disks, CD-ROM disks and DVD-ROM disks, and all forms of non-volatile digital and / or quantum memories, media, and memory devices including quantum systems such as trapped atoms or electrons. Quantum memory is understood to be a device that can store quantum data with high fidelity and efficiency for a long time, for example, where light is used for transmission and matter is used to store and preserve quantum features of quantum data such as superposition or quantum coherence, i.e., a light-matter interface.

[0174] The control of the various systems described herein, or portions thereof, can be implemented in a computer program product stored on one or more non-transitory machine-readable storage media and including instructions executable on one or more processing devices. The systems described herein, or portions thereof, can be implemented, respectively, as an apparatus, method, or system that includes one or more processing devices and a memory for storing executable instructions for performing the operations described herein.

[0175] This specification includes many details of particular implementations, which should not be considered as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. The particular features described herein in connection with separate implementations may be implemented in combination in a single implementation. Conversely, the various features described in connection with a single implementation may be implemented separately in multiple implementations or in any suitable partial combination. Further, features may be described above as acting in certain combinations and even initially claimed as such, but one or more features of the claimed combination may in some cases be capable of being excised from the combination, and the claimed combination may be directed to a partial combination or a variation of a partial combination.

[0176] Similarly, while operations are shown in the figures in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in a sequential order, or that all of the shown operations be performed, to achieve the desired result. In certain circumstances, multitasking and parallel processing may be advantageous. Further, the separation of various system modules and components in the above-described implementations should not be understood as necessarily requiring such separation in all implementations, and it should be understood that the described program components and systems may generally be integrated together in a single software product or packaged into multiple software products.

[0177] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims may be performed in a different order and still achieve the desired result. As one example, the processes shown in the accompanying figures do not necessarily require the particular order or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Explanation of Symbols

[0178] 100 Graph 200 Entangled Quantum Adversarial Generation Network (EQ-GAN) 202 True Data State Generator 204 Generator Network 206 Discriminator Network 208 True Data State 210 False Data State 212 Fidelity 214 Entangled Operation 216 Target Quantum State 300 Process 400 Circuit Diagram 402a~c Quantum State, Auxiliary Qubit 404 First Hadamard Gate 406 Unitary Operator 408 Second Hadamard Gate 410 Measurement Operation 412 Discriminator Output 500 Circuit Diagram 502 First 3-Qubit State, First State 504 Second 3-Qubit State, Second State 506 Exact Swap Test 508 First Hadamard Gate 510 Auxiliary Qubit 512 CSWAP Gate, Controlled-Swap Operation 514 Second Hadamard Gate 516 Measurement Operation 518 Swap Test 520 CNOT Gate 522 Hadamard Gate 524 Measurement Operation 526 Toffoli Gate 600 Graph 700 First Graph 750 Second Graph 1200 System 1202 Quantum Computing Device 1204 Control and Measurement System 1206 Qubit 1208 Coupler 1210 Control Signal 1212 Readout Signal 1214 Classical Processor 1216 Memory 1218 I / O Unit 1250 Classical Processor 1252 Qubit Assembly

Claims

1. A method for training a quantum adversarial generation network to learn a target quantum state, comprising: iteratively adjusting the parameters of the quantum adversarial generation network until the value of the quantum adversarial generation network loss function converges, each iteration comprising: performing a entanglement operation on the discriminator network input to measure the fidelity of the discriminator network input for the discriminator network of the quantum adversarial generation network, wherein the discriminator network input includes the target quantum state and a first quantum state output from a generator network of the quantum adversarial generation network, the first quantum state approximating the target quantum state; and performing a minimax optimization of the quantum adversarial generation network loss function to update the parameters of the quantum adversarial generation network, wherein the quantum adversarial generation network loss function depends on the measured fidelity of the discriminator network input. A method comprising the above steps.

2. The method according to claim 1, wherein the value of the quantum adversarial generation network loss function converges to a Nash equilibrium.

3. Each iteration further comprises: processing an initial quantum state by the generator network to output the first quantum state, the processing step including applying a first quantum circuit to the initial quantum state, i) the first quantum circuit being a parameterized quantum circuit, the first quantum circuit parameters constituting the parameters of the generator network included in the parameters of the quantum adversarial generation network. The method according to claim 1 or claim 2.

4. The method according to claim 3, wherein the first quantum circuit has a circuit depth smaller than that of the quantum circuit used to generate the target quantum state.

5. The method according to any one of claims 1 to 4, wherein the entanglement operation includes a parameterized entanglement operation approximating a swap test.

6. The method according to any one of claims 1 to 5, wherein the entanglement operation includes a swap test without an auxiliary qubit.

7. The swap test without an auxiliary qubit approximates an exact swap test and includes a second quantum circuit, the second quantum circuit being a parameterized quantum circuit. The method according to claim 6, wherein the second quantum circuit parameter constitutes the parameter of the discriminator network included in the parameter of the quantum adversarial generation network.

8. The method according to any one of claims 1 to 7, wherein the quantum adversarial generation network loss function includes one minus the measured fidelity of the discriminator network input.

9. The step of performing the min-max optimization of the quantum adversarial generation network loss function comprises: fixing the generator network parameters to the values determined in the previous iteration and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameters to determine the updated values of the discriminator network parameters for the iteration; and fixing the discriminator network parameters to the updated values of the discriminator network parameters for the iteration and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the updated values of the generator network parameters for the iteration The method according to any one of claims 1 to 8.

10. The step of performing the min-max optimization of the quantum adversarial generation network loss function comprises: fixing the discriminator network parameters to values corresponding to a complete swap test and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the first updated values of the generator network parameters for the iteration; fixing the generator network parameters to the first updated values and maximizing the quantum adversarial generation network loss function with respect to the discriminator network parameters to determine the updated values of the discriminator network parameters for the iteration; fixing the discriminator network parameters to the updated values of the discriminator network parameters for the iteration and minimizing the quantum adversarial generation network loss function with respect to the generator network parameters to determine the updated values of the generator network parameters for the iteration The method according to any one of claims 1 to 9.

11. The target quantum state includes a superposition state. The method according to any one of claims 1 to 10, further comprising the step of generating the target quantum state by the generator network according to the trained generator network parameters to approximate a quantum random access memory.

12. The method according to claim 11, further comprising the step of training a quantum neural network using the generated target quantum state.

13. The step of iteratively adjusting the parameters of the quantum adversarial generation network until the value of the quantum adversarial generation network loss function converges generates the trained generator network parameters and discriminator network parameters. The method according to any one of claims 1 to 12, further comprising the step of generating the target quantum state by the generator network according to the trained generator network parameters.

14. The step of performing a minimax optimization of the quantum adversarial generation network loss function to update the parameters of the quantum adversarial generation network includes the step of performing multiple circuit evaluations to calculate the gradient of the parameters of the quantum adversarial generation network. The method according to any one of claims 1 to 13.

15. A quantum adversarial generation network system implemented by one or more quantum computers, wherein the quantum adversarial generation network A discriminator network configured to perform an entanglement operation on the discriminator network input to measure the fidelity of the discriminator network input, The discriminator network input includes a target quantum state and a first quantum state output from a generator network included in the quantum adversarial generation network system, The quantum adversarial generation network system including a discriminator network, wherein the first quantum state approximates the target quantum state.

16. The quantum adversarial generation network system according to claim 15, wherein the entanglement operation includes a parameterized entanglement operation approximating a swap test.

17. The quantum adversarial generation network system according to claim 15 or claim 16, wherein the entanglement operation includes a swap test without ancilla.

18. The auxiliary-free swap test approximates a strict swap test and includes a second quantum circuit, the second quantum circuit being a parameterized quantum circuit, The quantum adversarial generation network system according to claim 17, wherein a second quantum circuit parameter constitutes a discriminator network parameter included in the parameters of the quantum adversarial generation network. **Claim 19** The quantum adversarial generation network system according to any one of claims 15 to 18, further comprising a generator network configured to apply a first quantum circuit to an initial quantum state to output the first quantum state. **Claim 20** the target quantum state including a superposition state, The quantum adversarial generation network system according to any one of claims 15 to 19, wherein the generator network generates the target quantum state to approximate a quantum random access memory according to trained generator network parameters.

Images