Quantum generative adversarial network with provable convergence

Abstract

Methods and 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 iteratively adjusting parameters of the QGAN until a value of a QGAN loss function converges, where each iteration includes performing an entangling operation on a discriminator network input to a discriminator network in the QGAN to measure a fidelity of the discriminator network input, where the discriminator network input includes the target quantum state and a first quantum state output from a generator network in the QGAN, where the first quantum state approximates the target quantum state, and performing a minimax optimization of the QGAN loss function to update the QGAN parameters, where the QGAN loss function depends on the measured fidelity of the discriminator network input.

Description

Background Technology

[0001] This manual covers quantum computing and generative adversarial networks.

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

[0003] Generative adversarial networks (GANs) are a form of generative machine learning that have achieved state-of-the-art performance in a variety of high-dimensional and complex tasks, including realistic image generation, super-resolution, and molecular synthesis. Given only the underlying data distribution p... data (x) The training dataset S = {x} i By accessing the domain}, GANs can generate realistic examples outside of S. Some probability distributions are often difficult to sample, therefore learning an arbitrary distribution p data The precise representation of (x) can benefit from access to quantum computing resources. Summary of the Invention

[0004] This specification describes quantum generative adversarial networks with provably convergent properties.

[0005] Typically, an innovative aspect of the subject matter described in this specification can be implemented in a method for training a quantum generative adversarial network (GAN) to learn a target quantum state, the method comprising: iteratively adjusting the parameters of the GAN until the value of the GAN loss function converges, wherein each iteration comprises: performing an entanglement operation on the discriminator network input of the discriminator network in the GAN to measure the fidelity of the discriminator network input, wherein the discriminator network input includes the target quantum state and a first quantum state output from the generator network in the GAN, wherein the first quantum state approximates the target quantum state; and performing minimax optimization of the GAN loss function to update the parameters of the GAN, wherein the GAN loss function depends on the fidelity of the measured discriminator network input.

[0006] Other embodiments of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each computer system, apparatus, and computer program configured to perform actions of the method. One or more classical and / or quantum computer systems may be configured to perform specific operations or actions by installing software, firmware, hardware, or combinations thereof on the system, which in operation causes the system to perform actions. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform actions.

[0007] The foregoing and other implementations may each optionally include one or more of the following features, individually or in combination. In some implementations, the value of the quantum generative adversarial network loss converges to a Nash equilibrium.

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

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

[0010] In some implementations, the entanglement operation includes a parameterized entanglement operation with an approximate swap test.

[0011] In some implementations, the entanglement operation includes an unassisted swap test.

[0012] In some implementations, the unassisted exchange test approximates the exact exchange test and includes a second quantum circuit, wherein the second quantum circuit is a parameterized quantum circuit, and the parameters of the second quantum circuit constitute the discriminator network parameters included in the parameters of the quantum generative adversarial network.

[0013] In some implementations, the loss function of a quantum generative adversarial network includes 1 minus the measurement fidelity of the discriminator network input.

[0014] In some implementations, performing minimax optimization of the quantum generative adversarial network loss function includes: fixing the generator network parameters to values ​​determined in previous iterations and maximizing the quantum generative adversarial network loss function relative to the discriminator network parameters to determine updated values ​​for the discriminator network parameters for that iteration; and fixing the discriminator network parameters to the updated values ​​for the discriminator network parameters for that iteration and minimizing the quantum generative adversarial network loss function relative to the generator network parameters to determine updated values ​​for the generator network parameters for that iteration.

[0015] In some implementations, the method further includes fixing the discriminator network parameters to values ​​corresponding to a perfect swap test and minimizing a quantum generative adversarial network loss function relative to the generator network parameters to determine initial update values ​​for the generator network parameters for that iteration; fixing the generator network parameters to the initial update values ​​and maximizing the quantum generative adversarial network loss function relative to the discriminator network parameters to determine update values ​​for the discriminator network parameters for that iteration; and fixing the discriminator network parameters to the update values ​​for the discriminator network parameters for that iteration and minimizing the quantum generative adversarial network loss function relative to the generator network parameters to determine update values ​​for the generator network parameters for that iteration.

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

[0017] In some implementations, the method also includes using the generated target quantum state to train a quantum neural network.

[0018] In some implementations, the parameters of the quantum generative adversarial network are iteratively adjusted until the value of the quantum generative adversarial network loss function converges to generate trained generator network and discriminator network parameters, and the method further includes using the generator network and generating a target state based on the trained generator network parameters.

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

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

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

[0022] The currently described entangled quantum generative adversarial network (EQ-GAN) has been proven to converge to the globally optimal Nash equilibrium and to converge to problem instances where conventional quantum generative adversarial networks (QGAN) fail.

[0023] Furthermore, the task of learning quantum circuits to generate unknown quantum states can also be solved using a fully supervised approach. Instead of training the discriminator adversarially to distinguish between fake and real data, the discriminator can be frozen to perform an exact swap test, thus measuring the state fidelity between real and fake data. While this will reproduce the original state without noise, gate errors in the discriminator implementation will cause convergence to an incorrect optimum. Results show that the adversarial approach of EQ-GAN is more robust to such errors than simpler supervised learning methods. Since training quantum machine learning models can require significant time to compute gradients on current quantum hardware, resilience to gate error drift during training 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 with classical data are presented. Most quantum machine learning algorithms, promising exponential speedup compared to their classical counterparts, require quantum random access memory (QRAM). By learning shallow quantum circuits to generate superpositions of classical data, EQ-GAN can be used to create an approximate QRAM. It can be shown that applying this QRAM to quantum neural networks increases the accuracy of quantum neural network classification with the same amount of training time as using classical datasets. Once trained, the quantum neural network can be easily inverted to provide interpretability of its classification process. EQ-GAN offers a new paradigm for loading classical data into quantum states prepared by shallow quantum circuits through variational circuit optimization.

[0025] Details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the specification, drawings, and claims. Attached Figure Description

[0026] Figure 1 The curves show the performance of conventional quantum generative adversarial networks.

[0027] Figure 2 Describe an example of an entangled quantum generative adversarial network.

[0028] Figure 3 This is a flowchart of an example process for training a quantum generative adversarial network to learn a target quantum state. The quantum generative adversarial network includes a generator network and a discriminator network.

[0029] Figure 4A The circuit diagram shown illustrates an example EQ-GAN discriminator architecture.

[0030] Figure 4B An example representation of the unitary operator is shown.

[0031] Figure 5 The circuit diagram for an unassisted exchange test between two 3-qubit states is shown.

[0032] Figure 6 The diagram shows a curve comparing QGAN with the currently described EQ-GAN, which learns quantum states.

[0033] Figure 7 The diagram shows a first curve comparing EQ-GAN and a supervised learner implemented on a simulated quantum device, and a second curve comparing EQ-GAN and a supervised learner implemented on a physical quantum device.

[0034] Figure 8 Two variational QRAM fitting methods are shown for generating peak values.

[0035] Figure 9 The bimodal total dataset and variational QRAM of the training dataset are shown.

[0036] Figure 10 An example quantum neural network architecture and its corresponding layout on a physical device are shown.

[0037] Figure 11 The decomposition of the rank-4 biqubit entanglement gate is shown.

[0038] Figure 12 An example system is shown.

[0039] The same reference numerals and names in the various figures indicate the same elements. Detailed Implementation

[0040] Generative Adversarial Networks (GANs) include parameterized generator networks G(θ) g (z) and parameterized discriminator network D(θ)d The generator transforms the vector sampled from the input distribution z ~ p0(z) into data samples G(θ). g This transforms p0(z) into a new distribution p of dummy data. g (z). The discriminator takes an input sample x and gives the probability D(θ) that the sample is real (from the data) or fake (from the generator network). d Training corresponds to a minimax optimization problem, where an alternation is achieved between improving the discriminator's ability to distinguish between real and fake samples and improving the generator's ability to deceive the discriminator. For example, The cost function V is solved as given by the following equation (1).

[0041]

[0042] In equation (1), θ g θ represents the generator network parameters. d p represents the discriminator network parameters. data (z) represents the true data distribution, and p0(z) represents the input distribution.

[0043] If G and D have sufficient capacity, such as a space close to that of any function, then it is proven that the global optimum of this minimax game exists and uniquely corresponds to p. g (x)=p data (x). While multilayer perceptrons can be used to parameterize D and G, the dimension of the function space can also be increased by replacing classical neural networks with quantum neural networks. In the most general case, classical data can be represented by the density matrix σ = ∑ i p i |ψ i ><ψ i | indicates that p i ∈[0,1] denotes a positively bounded real number, and |ψ i > is an orthogonal basis state. In the first proposal of Quantum GAN (QGAN), the generator network outputs quantum states from the initial state ρ0. The quantum circuit U is defined as follows. The discriminator acquires either true data σ or false data ρ and performs a positive operator value measurement (POVM) to return either a true data operator T or a false data operator F, where ||T||1, ||F||1≤1. Therefore, any state ρ... in The probability of a given value being true is given by the following formula:

[0044] D(θ d , ρ in )=Tr[Tρ in (2)

[0045] QGAN solves the minimum-maximum game, for example, as given by the following equation (3).

[0046]

[0047] Since the set of positive operators with a norm less than or equal to 1 is convex and compact, gradient descent can be used to optimize the discriminator's measurement.

[0048] The optimal discriminator measurement is given by the Helstrom measurement, where the operator P + (σ-ρ) and 1-P + (σ-ρ) distinguishes between the positive and negative parts of σ-ρ. That is, given a strictly positive proof value... and strictly negative proof value (with corresponding eigenstates) and ) extension

[0049]

[0050] The best discriminator will select and F = 1 - P + (σ-ρ). To obtain from D(θ) d ,σ)-D(θ) d ,ρ(θ) g ))=Tr[Tσ]-Tr[Tρ(θ g To achieve Nash equilibrium, the generator must modify θ if θ > 0. g , such that Tr[Tρ(θ) g The value will increase. Although some methods propose updating ρ→ρ′=ρ+α(σ-ρ) for α>0 by minimizing equation (3), this strategy does not produce Nash equilibrium. Evaluating the trace with the T operator aligns the generated data only with the orthogonal projection of σ-ρ. This ultimately leads to mode collapse, as shown in the example below.

[0051] Consider the true data state σ and the generator initialized in state ρ, where each state is defined by the following equation:

[0052]

[0053]

[0054] Where, σ x Let x and σ represent the Pauli operators. y Represent the Pauli operator y. By decomposing σ - ρ = σ y / 2 maximizes the relationship between equation (3) and the Helstrom measurement, and the discriminator will take Optimization is performed in the space of the density matrix. The generator rotates ρ to be parallel to T, and also gives... In the next iteration, the discriminator attempts to perform a new Helstrom measurement to distinguish σ from ρ′, but this results in T′ = P. + (σ-ρ′)=ρ. When the generator is realigned to match the new measurement operator, ρ″=ρ. It is now directly apparent that if QGAN is trained to completely solve the minimax optimization problem at each iteration, it will never converge. Instead, it will always oscillate between states ρ and ρ′, neither of which is a Nash equilibrium of the minimax game in terms of QGAN performance under this mode collapse.

[0055] Figure 1 Curve 100 illustrates the performance of traditional QGAN learning the state defined in Equation (5), where the initialization is given by Equation (6). The x-axis represents the number of training episodes. The y-axis represents the loss. Curve 100 shows that mode collapse manifests as oscillations in the generator and discriminator loss, failing to converge to the global optimum.

[0056] More generally, consider the oscillations between finite sets of states. Let the function T σ (ρ)=P + (σ-ρ) represents the best Helstrom measurement obtained from the positive part of the spectral decomposition of σ-ρ in equation (4). if If it is a k-fold combination with itself, then there exist some k>1 such that T (k) =ρ is sufficient to ensure oscillations between k states. For a system with n qubits, this can be achieved by preparing the target state and the initial state separated by an angle π / 3 on a generalized Bloch sphere.

[0057] While this problem persists only if the discriminator in a QGAN is allowed to converge to the Helstrom measurement during training, it can make the QGAN architecture more sensitive to the choice of hyperparameters, particularly the learning rate and the number of epochs per iteration for training the discriminator and generator. With a fully trained QGAN discriminator and generator, mode collapse results in ρ and ρ′ sharing 3 / 4 of the fidelity, thus remaining constant throughout the oscillation. However, even with a standard learning rate and only one epoch per iteration, QGAN oscillates at the start of training. Unstable training is difficult to overcome even in classical GAN ​​architectures, so understanding how to prevent this non-convergent progress is important for both quantum and classical machine learning.

[0058] This specification describes a novel quantum GAN that does not experience the aforementioned mode collapse, thus providing a more robust QGAN architecture. The novel quantum GAN is an entangled QGAN (referred to as EQ-GAN in this paper), which, instead of feeding the discriminator real or fake data, entangles both real and fake data.

[0059] Example operating environment

[0060] Figure 2 This is a block diagram of the Entangled Quantum Generative Adversarial Network (EQ-GAN)200.

[0061] The EQ-GAN200 includes a true data state generator 202. The true data state generator 202 includes quantum hardware configured to generate a target quantum state (e.g., true data state 208). In some implementations, the true data state generator 202 can prepare the target quantum state by applying quantum circuitry to an initial quantum state. In some cases, such as when the target quantum state is a superposition of classical data, the quantum circuitry required to generate a particular target quantum state may include quantum logic gates that are expensive to implement and / or may have large circuit depths. Therefore, generating a large number of target quantum states may be inefficient or infeasible.

[0062] The EQ-GAN200 also includes a generator network 204. Generator network 204 is configured to generate quantum states that approximate the target quantum state, such as pseudo-data state 210. For example, as described in more detail below, the generator network may include quantum computing hardware configured to apply parameterized quantum circuitry to the initial quantum state to output a quantum state that approximates the target quantum state. Compared to the quantum circuitry implemented by the true data state generator 202, the parameterized quantum circuitry can have a lower circuit depth to produce an accurate target quantum state. Therefore, training generator network 204 by adjusting the parameters of the parameterized quantum circuitry until the value of the EQ-GAN loss function converges allows generator network 204 to produce an accurate approximation of the target quantum state at a lower computational cost. (See below for reference...) Figure 3-11 A more detailed description of the example operations performed by generator network 204 is provided below. (Refer to the following...) Figure 12 The example hardware included in generator network 204 is described in more detail.

[0063] The EQ-GAN 200 also includes a discriminator network 206. The discriminator network 206 is configured to receive discriminator network inputs and perform an entanglement operation 214 on the discriminator network inputs to measure the fidelity 212 of the discriminator network inputs. The discriminator network inputs include a true data state 208 obtained from the true data state generator 202 and a false data state 210 output from the 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 of an approximate swap test. In some embodiments, the entanglement operation requires auxiliary qubits. In other embodiments, the entanglement operation is an unassisted approximation of the swap test. In either case, the entanglement operation can be implemented by applying parameterized quantum circuitry. (Refer to the following...) Figure 3-11 A more detailed description of the example operations performed by the discriminator network 206 is provided below. (Refer to the following...) Figure 12 A more detailed description of the example hardware included in the discriminator network 206 is provided.

[0064] EQ-GAN 200 can be trained so that generator network 204 learns an improved approximation of the target quantum state using quantum circuitry. During training, the generator network 204 generates fake data states and the discriminator network 206 learns fidelity measurements adversarially until a convergence criterion is met. Once trained, generator network 204 can be used to generate approximations of the target quantum state 216 based on the trained generator network parameters, for example, to approximate a quantum random access memory. (See below for reference.) Figure 3 Describe an example process for training an EQ-GAN to learn a target quantum state.

[0065] Example process for training EQ-GAN

[0066] Figure 3 This is a flowchart of an example process 300 for training a quantum generative adversarial network to learn a target quantum state. The quantum generative adversarial network includes a generator network and a discriminator network. For convenience, process 300 will be described as being executed by quantum hardware communicating with control electronics located at one or more locations. For example, a quantum hardware appropriately programmed according to this specification... Figure 2 System 200 can execute process 300.

[0067] The parameters θ of the quantum generative adversarial network are iteratively adjusted systematically. g ,θ d This continues until the value of the quantum generative adversarial network loss function converges. The quantum generative adversarial network loss function is described below with reference to equation (8).

[0068] In each iteration, the generator network processes the initial quantum state ρ0 to output the quantum state ρ (step 302). This processing may include applying a first quantum circuit ρ to the initial quantum state, wherein the first quantum circuit is a parameterized quantum circuit and includes parameters θ constituting the generator network. g The parameters. That is, the quantum state can be determined by... Given, where U(θ) g ) represents the parameterized first quantum circuit, θ g Let θ represent the generator network parameters, and ρ0 represent the initial quantum state. The quantum state ρ approximates the target quantum state, and the generator network parameters θ are iteratively adjusted. g Make the first quantum circuit U(θ) g It can generate a better approximation of the target quantum state. Processing can be performed using quantum hardware.

[0069] At each iteration, the discriminator network performs an entanglement operation on its input to measure the fidelity of the 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 analogous to the discriminator in a classical GAN. Instead of evaluating fake or real data separately, the discriminator is always given access to the real data σ and the input state ρ. in Perform fidelity measurement As shown in equation (7) below.

[0070]

[0071] This allows the quantum generative adversarial loss function to converge to a Nash equilibrium. Entanglement operations can be performed using quantum hardware.

[0072] The entanglement operation performed by the discriminator network is a parameterized entanglement operation approximating the swap test. The swap test is a process in quantum computing used to check how different two quantum states are. The swap test requires an auxiliary qubit, for example, initialized in a zero state, and is performed by repeatedly applying a Hadamard gate to the auxiliary qubit, applying a CSWAP gate (also called a controlled swap gate or Fredkin gate) to the qubit pair from the first and second quantum states, applying a Hadamard gate to the auxiliary qubit, and measuring the auxiliary qubit, for example, on a Z-based basis, to determine how different the quantum states are.

[0073] In some implementations, the discriminator network D σ (θ d , ρ inThis could be a parameterized quantum circuit with auxiliary qubits—as in the case of a precise swap test—where the circuit parameters constitute the discriminator network parameters. If parameters exist to implement a precise swap test... Right now Then D σ The discriminator is fully expressed to achieve the optimal discriminator during optimization. However, since the conventional exchange test across two n-qubit states requires two qubit gates spanning 2n qubits, its implementation on quantum devices with local connectivity incurs excessive overhead in terms of circuit depth. Therefore, in some implementations, the discriminator network can be a parameterized circuit with auxiliary qubits having an approximate exchange test.

[0074] Figure 4A This is a circuit diagram 400 illustrating an example discriminator network architecture. The example 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., a zero state). The second quantum state 402b is the output ρ(θ) of the generator network. g For example, pseudo data. The third quantum state 402c is the target quantum state σ of one or more qubits, for example, true data.

[0075] 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 generator network outputs ρ(θ). g The unitary operator 406 depends on a set of discriminator parameters θ and the target quantum state σ. d And an approximate commutation test is performed. The unitary operator 406 can represent a sequence of quantum logic gates, and the quantum logic gates included in this sequence can vary based on the sizes of the second and third quantum states and the specific hardware implementation. For example, when 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 include single-qubit rotation gates, S-gates, T-gates, Hadamard gates, Pauli-X gates, and CZ gates. Figure 4B The circuit representation of example unitary operator 406 is shown below. Figure 4B In the diagram, X1-X7 represent the free parameters to be trained.

[0076] The discriminator network also applies a second Hadamard gate 408 to an auxiliary qubit 402a and uses a measurement operation 410 to measure the auxiliary qubit to obtain a discriminator output 412 representing the difference between the second quantum state 402b and the third quantum state 402c.

[0077] To further simplify the physical implementation of the discriminator network, in some implementations, the discriminator network D... σ (θd , ρ in The discriminator network can be a parameterized circuit that does not include auxiliary qubits. Conversely, the discriminator network can be a parameterized circuit that performs a destructive unaided (unaided qubit) approximation on the exchange test.

[0078] For example, for quantum devices with planar connectivity, the CNOT gate can be decomposed into The operation utilizes the native CZ gate. The CZ gate has an unstable error that can be efficiently modeled by rotating an unknown angle on any qubit using Z rotations. The currently described EQ-GAN formal system overcomes the single-qubit phase error by directly applying the RZ(θ) gate after each CZ operation. During adversarial training, gradient descent is used to optimize the free angle θ to mitigate the two-qubit gate error. Due to the convergence properties provided by the generative adversarial framework, the discriminator can be proven to converge to the possible optimal state discriminator. This triggers early stopping (e.g., when the discriminator loss indicates that the optimal state discriminator has been achieved) Figure 7 (As shown).

[0079] Figure 5 This is a circuit diagram 500 for an unassisted approximate swap test between a first 3-qubit state 502 and a second 3-qubit state 504. The left-hand side of 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, three CSWAP gates 512 are applied to the qubit pairs of the first 3-qubit states 502 and 504, for example, a first CSWAP gate is applied to the first qubit in the first state 502 and the first qubit in the second state 504, a second CSWAP gate is applied to the second qubit in the first state 502 and the second qubit in the second state 504, and a third CSWAP gate is applied to the third qubit in the first state 502 and the third qubit in the second state 504, wherein the auxiliary qubit 510 acts as the controller 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 a measurement result for the auxiliary qubit.

[0080] The right-hand side of circuit diagram 500 shows an alternative implementation of the swap test 518. By rewriting the controlled swap operation 512 as a CNOT gate 520 applied to the corresponding qubit in the first state 502, where the corresponding qubit in the second state 504 is used as control for each CNOT gate, a Hadamard gate 522 is applied to each qubit in the second state 504, a measurement operation 524 is performed on each qubit in the first state 502 and the second state 504, a Tofoli gate 526 is applied to the auxiliary classical bit, where each Tofoli gate uses the corresponding qubit in the first state 502 and the corresponding qubit in the second state 514 as control, and the computational basis operation is replaced with classical post-processing, the swap test can be performed with the auxiliary classical bit without the auxiliary qubit (hence the term "unassisted" swap test).

[0081] return Figure 3 The system performs minimax optimization of the quantum generative adversarial network loss function to update the parameters of the quantum generative adversarial network (step 306). Minimax optimization can be performed using one or more classical processors. The quantum generative adversarial network loss function depends on the discriminator network output minus the discriminator network input D. σ (θ d ,ρ(θ) g The fidelity of the measurement is equal to 1 minus the fidelity of the measurement of the discriminator network input, as given in equation (8) below.

[0082]

[0083] Performing minimax optimization of the loss function of a quantum generative adversarial network involves adjusting the generator network parameters θ. g Fixed to the value determined in the previous iteration (or the initial value if this is the first iteration), and relative to the discriminator network parameters θ. d Maximize the quantum generative adversarial network loss function to determine the updated values ​​of the discriminator network parameters for this iteration, and fix the discriminator network parameters to the discriminator network parameters θ for this iteration. d The updated values ​​and relative to the generator network parameters θ g Minimize the quantum generative adversarial network loss function to determine the updated values ​​of the generator network parameters for this iteration. In other words, the EQGAN architecture adversarially optimizes the state ρ(θ). g Generation and fidelity measurement of D σ The learning.

[0084] The parameters of the quantum generative adversarial network (GAN) are iteratively adjusted until the GAN loss function converges, as described above with reference to steps 302-306, to generate trained generator network parameters and discriminator network parameters, defining the trained generator network and the trained discriminator network. Once trained, the generator network can generate an accurate approximation of the target state based on the trained generator network parameters, for example, as part of the QRAM described below.

[0085] We now show that there exists a unique Nash equilibrium at the desired location. By definition, 0 ≤ D σ (θ d ,ρ(θ) g ))≤1 is Figure 4A or Figure 5 The circuit shown measures the probability of state |1> at its end. If the discriminator implements the identity transformation U(θ)... d If )=II, then the probability of observing state |1> is zero. In the first step of discriminator maximization, the discriminator performs a nontrivial entanglement operation on the generator output and the real data. Furthermore, for U(θ) d Given a switch-test circuit ansatz, the maximum value used to distinguish between two arbitrary states is uniquely achieved through a perfect switch-test angle. While the discriminator may not choose the switch test, the next step is to minimize D from the generator side. σ (θ d ,ρ(θ) g If the discriminator does not implement the exchange test, the generator can choose a new arbitrary state that will be poorly distinguished by the discriminator because it does not use a fidelity comparison. Ultimately, the generator cannot improve if and only if the discriminator uses the exchange test, in which case the only minimum lies at ρ. in =σ.

[0086] The circuit parameterization U(θ) can be selected based on various factors. d These factors include the type of device used to implement the discriminator network, the available connectivity within the device, and the types of gates that can be efficiently implemented by the device. For example, for recent quantum devices with planar connectivity, fixed gates or two-qubit entangled gates can be efficiently implemented, and thus circuit parameterization can be formed.

[0087] While poorly chosen circuit parameterization can result in a non-convex loss function shape and thus be difficult to optimize via gradient descent, this is a problem shared with QGANs due to the difficulty in representing arbitrary unary circuits as shallow quantum circuits; similarly, non-convexity in classical GANs often hinders convergence. However, the EQ-GAN architecture successfully converges to problem instances that cannot be reached by a fully trained and properly parameterized QGAN. Figure 6Curve 600 shows a comparison between QGAN and the currently described EQ-GAN, which learns the quantum states given by equation (5). The x-axis represents the number of iterations. The y-axis represents the overlap with the data states. Curve 600 shows that while QGAN oscillates infinitely between two states of equal fidelity (3 / 4), EQ-GAN converges rapidly to full fidelity.

[0088] Learning to suppress errors

[0089] Compared to more direct supervised learning methods used to learn unknown quantum states, EQ-GAN achieves improved robustness against gate errors. Instead of training the parameterized swap test used as the discriminator in EQ-GAN adversarially, a perfect swap test can be applied in each iteration by freezing the discriminator. This can also lead to the generator circuit converging to real data, as the swap test ensures a unique global optimum.

[0090] However, in the presence of gate errors in the swap test, this unique global optimum will be offset from the true data. Since the exact parameterization of EQ-GAN for the perfect swap test is unknown, an appropriate ansatz can be learned to correct for the coherent errors observed on recent quantum hardware. In particular, the gate parameters in the two-qubit entangled gate (such as the conditional Z-phase, single-qubit Z-phase, and swap angle) can drift and oscillate on a timescale of O(10) minutes. This unknown system- and time-dependent coherent error presents a significant challenge for applications in quantum machine learning, where gradient computation and updates require numerous measurements.

[0091] By incorporating additional single-qubit Z-phase compensation, the large deviations in Z-rotation angles for both single-qubit and two-qubit systems can be significantly mitigated. Recently, the effectiveness and importance of this systematic error mitigation have been demonstrated in achieving state-of-the-art accuracy in energy estimation of fermion molecules. The adversarial learning of EQ-GANs provides a useful paradigm for learning discriminator circuits that most closely approximate real-world exchange tests, and can be broadly applied to improve the fidelity of other recent quantum algorithms.

[0092] Assume the unitary matrix of the adversarial discriminant is given by U(θ) d ) is given, where This corresponds to the perfect swap test in the absence of noise. Given a trace-preserving, all-positive noise channel ε, a new unitary operation is used. To replace the discriminator. Although the supervised method will apply the... The given approximate commutation test, but if parameters exist... Make Adversarial exchange tests will generally perform better. Because the discriminator defines a loss form optimized by the generator, if the noise unitary matrix... The parameterization is generally sufficient to mitigate the error, then the ρ(θ) generated by EQ-GAN g It can converge to a state that is closer to σ than the state that the supervision method could possibly reach.

[0093] Since the discriminator must converge to the exchange test at the optimal Nash equilibrium, convergence can be heuristically improved through two training phases in the presence of noise. In the first phase, the discriminator is frozen with the parameters of the perfect exchange test, although the unitary matrix... This may involve imperfect swapping tests; the generator is trained until the loss converges. In the second phase of training, the discriminator is allowed to vary adversarially against the generator, thereby finding the parameters. In the case of a gate error, this second stage can produce a unitary matrix that is closer to a real exchange test.

[0094] In other words, performing minimax optimization of the quantum generative adversarial network (GAN) loss function can include: fixing the discriminator network parameters to values ​​corresponding to perfect swap tests and minimizing the GAN loss function relative to the generator network parameters to determine the initial update values ​​of the generator network parameters for that iteration; fixing the generator network parameters to the initial update values ​​and maximizing the GAN loss function relative to the discriminator network parameters to determine the update values ​​of the discriminator network parameters for that iteration; and fixing the discriminator network parameters to the update values ​​of the discriminator network parameters for that iteration and minimizing the GAN loss function relative to the generator network parameters to determine the update values ​​of the generator network parameters for that iteration.

[0095] As an example, consider learning superposition states on a noisy quantum device. The task is as follows. Following the heuristic described above, EQ-GAN is trained using a frozen discriminator during the first half of training and adversarially during the second half. The discriminator is defined by a swap test, where a CZ gate provides the necessary two-qubit operation. However, to learn to correct for gate errors, the discriminator adversarially learns the angle of insertion of a single qubit Z-rotation directly after the CZ gate. Therefore, EQ-GAN achieves state overlap that is significantly better than the state overlap in a perfect swap test.

[0096] Table I shows the average error after running EQ-GAN and the supervised learner multiple times on the experimental equipment.

[0097] Supervised learners <![CDATA[(2.4±0.5)×10 -4 ]]> EQ-GAN <![CDATA[(0.4±0.3)×10 -4 ]]>

[0098] Table I

[0099] Table I compares EQ-GAN and supervised learners on quantum devices with 50 qubits, CZ gates, and arbitrary single-qubit gates. The comparison shows that the error of EQ-GAN (i.e., 1-state fidelity relative to real data) is significantly lower than that of the supervised learner, demonstrating successful adversarial training with error suppression swapping tests. Uncertainty is shown as two standard deviations.

[0100] Figure 7 The first curve 700, comparing EQ-GAN with a supervised learner implemented on a simulated quantum device, and the second curve 750, comparing EQ-GAN with a supervised learner implemented on a physical quantum device, are shown. In both graphs, the x-axis represents the number of iterations, and the y-axis represents the state fidelity with respect to real data. In the simulation, normal distribution noise on individual qubit rotations is imposed with a systematic bias far from zero, causing the discriminator of the supervised learner to be forced to converge to an incorrect state. Experiments confirm that by learning to correct this error with additional individual qubit rotations, EQ-GAN converges to a higher state overlap. The converged EQ-GAN (dashed line) is determined by iterations that allow the discriminator loss to reach an extremum.

[0101] QRAM Applications

[0102] Many quantum machine learning applications require quantum random access memory (QRAM) to load data superimposed. However, loading arbitrary states may require noisy controlled rotations, and preparing a superposition of an arbitrary set of n states requires at most O(n) operations. Given a suitable ansatz, EQ-GAN can be used to learn states that are approximately equivalent to a superposition of data. That is, the above reference Figure 3 The target quantum state can be a superposition state representing a superposition of data, and once trained, the generator network can be used to generate superposition states based on the trained generator network parameters to approximate QRAM. Quantum acceleration can be achieved if training EQ-GAN is computationally less costly than the number of calls required for QRAM in the context of another algorithm.

[0103] To demonstrate the variational QRAM, a dataset of two peaks sampled from different Gaussian distributions is used. While accurately encoding the empirical probability density function requires very deep circuitry and multiple control rotations, a shallow circuit fitting method that generates exponential peaks can be chosen. Figure 8 Two variational QRAM fitting methods are shown for generating peaks. Category 0 corresponds to the central peak, and category 1 corresponds to the offset peak. Once trained to approximate the empirical data distribution, the variational QRAM closely reproduces the original dataset. Figure 9The variational QRAM (N=60) is shown for the bimodal total dataset (sampled from a normal distribution) and the training dataset. The variational QRAM is obtained by training an EQ-GAN to generate states σ with shallow peaks ansatz, approximating an exact superposition of states σ. The training and test datasets (each N=60) are balanced between the two classes.

[0104] As proof of the principle of using this QRAM in a quantum machine learning context, a quantum neural network can be trained using the QRAM described above, and the hinge loss can be computed either by considering each data entry individually (encoded as a quantum circuit) or by considering each class individually (encoded as a superposition in a variational QRAM). Given the same number of circuit evaluations to compute the gradient, the superposition converges to better accuracy at the end of training, although using an approximate distribution, as shown in Table II below.

[0105] Precise sampling 53％±6％ Variational QRAM 69％±2％

[0106] Table II

[0107] Table II shows the test accuracy (N=60) of a quantum neural network (QNN) trained on all samples of the training dataset (N=60) for a single epoch or trained on a variational QRAM for an equal number of circuit evaluations. Although the QNN trained on the variational QRAM does not have direct access to the original dataset, its accuracy is evaluated on the original dataset. Uncertainty is shown as two standard deviations.

[0108] Empirical performance differences can be demonstrated between training a quantum neural network (QNN) using individual examples from classical datasets and training a QNN using a superposition of data obtained from a pre-trained EQ-GAN. Arbitrarily parameterized circuits with single-qubit gates and two-qubit gates can be used to construct QNN ansatz. Due to the planar connectivity of quantum devices with 50 qubits, CZ gates, and arbitrary single-qubit gates, Figure 10 The QNN shown can be implemented using four-qubit data states. Figure 10 An example quantum neural network architecture (left) and its corresponding layout on a quantum device (right) are shown. Figure 7 The circuit shown constructs a four-qubit data state and places it in the |data> state on the blue qubit. The qubit (orange) is read out and then executed. Figure 11 The parameterized two-qubit interaction is shown. To implement the rank-4 entanglement gate G given by the following equation, a native CZ two-qubit gate is used.

[0109]

[0110] It can be like Figure 11The decomposition is shown. Instead of using ZZ interactions as in some conventional proposals, any two-qubit entangled interaction can be freely chosen to construct the parameterized unitary matrix.

[0111] Figure 11 The decomposition of the biqubit entanglement gate G(θ) used in QNN ansatz is shown by equation (9).

[0112] QNNs can be trained in two ways—via sampling or via stacking. As mentioned above, the stacking method cannot use an exact stack of the training dataset. Instead, it can use a shallow approximation obtained through pre-training an EQ-GAN. For fair comparison, an equal number of queries to the quantum device are allowed. Therefore, for N = 60 examples, with 30 examples per class, training via sampling is performed for one epoch, where 60 corresponds to 60 iterations performed on the quantum device. However, training via stacking evaluates 30 stacks per class (since there are two classes), also accessing the quantum device in 60 iterations. Additionally, Bayesian optimization is used to adjust the different learning rates for the sampling and stacking methods. In the simulation, optimization is performed from 10 random parameter trials and 40 evaluations of the Gaussian process estimator. -4 Up to 10 -1 The Adam learning rate was used. For each parameter query, the QNN output was averaged over 10 trials to reduce any statistical fluctuations. The final learning rate was then evaluated over 50 trials (for samples of 10-10). -3.93 For a superposition of 10- 1.83 The QNN is used to obtain the final performance with the calculated standard deviation reported in Table II.

[0113] Figure 12 Example system 1200 is depicted for performing classical and quantum computing as described in this specification. Example system 1200 is an example of a system implemented as a classical and quantum computer program on one or more classical and quantum computing devices in one or more locations, wherein the systems, components and techniques described herein may be implemented.

[0114] Example system 1200 includes example quantum computing device 1202. According to some embodiments, quantum computing device 1202 can be used to perform the quantum computing operations described in this specification. 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 implementation of the inventions described and / or claimed in this document.

[0115] Example quantum computing device 1202 includes a qubit assembly 1252 and a control and measurement system 1204. The qubit assembly includes multiple qubits, such as qubit 1206, for performing algorithmic operations or quantum computing. Although Figure 12 The qubits shown are arranged in a rectangular array, but this is a schematic depiction and not intended to be limiting. The qubit assembly 1252 also includes adjustable coupling elements, such as coupler 1208, which allows interaction between the coupled qubits. Figure 12 In the schematic depiction, each qubit is tunably coupled to each of its four neighboring qubits by means of a corresponding coupling element. However, this is an example arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include tunable coupling between more than two qubits.

[0116] Each qubit can be a physical two-level quantum system or device with levels representing logical values ​​0 and 1. The specific physical implementation of multiple qubits and how they interact with each other depends on various factors, including the type of quantum computing device included in example system 1200 or the type of quantum computing the device is performing. For example, in an atomic quantum computer, qubits can be implemented via atoms, molecules, or solid-state quantum systems (e.g., hyperfine atomic states). As another example, in a superconducting quantum computer, qubits can be implemented via superconducting or semiconducting qubits (e.g., superconducting transmon states). As yet another example, in an NMR quantum computer, qubits can be implemented via nuclear spin states.

[0117] In some implementations, quantum computing can be performed by initializing qubits in a selected initial state and applying a sequence of unitary operators to the qubits. Applying unitary operators to the quantum state can include applying a corresponding sequence of quantum logic gates to the qubits. Example quantum logic gates include: single-qubit gates, such as Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z)), Hadamard gates, S-gates, and rotations; two-qubit gates, such as controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ)), controlled NOT gates (also referred to as CNOT), controlled swapping gates (also referred to as CSWAP); and gates involving three or more qubits (e.g., Tofoli gates). The quantum logic gates can be implemented by applying control signals 1210 generated by the control and measurement system 1204 to the qubits and couplers.

[0118] For example, in some implementations, the qubits in the qubit assembly 1252 may be frequency-tunable. In these examples, each qubit may have an associated operating frequency that can be adjusted by applying voltage pulses via one or more drive lines coupled to the qubit. Example operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency can place the qubit in a state where it does not interact strongly with other qubits and can be used to perform a single-qubit gate. As another example, where the qubits interact via couplers with fixed coupling, the qubits can be configured to interact with each other by setting their respective operating frequencies to a gate-related frequency that is detuned from their common interaction frequency. In other cases, for example, when the qubits interact via tunable couplers, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to achieve interaction between the qubits, and then interacting with each other by setting the respective operating frequencies of the qubits to a gate-related frequency that is detuned from their common interaction frequency. Such interactions can be performed to perform multi-qubit gates.

[0119] 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 light pulses in an atomic quantum computer system.

[0120] Quantum computing can be performed by measuring the state of a qubit using a corresponding control signal 1210, for example, using a quantum observable such as X or Z. 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 scheme of the quantum computing device and / or the qubit. For convenience, Figure 12 The control signal 1210 and readout signal 1212 shown are depicted as addressing only selected elements of the qubit assembly (i.e., the top and bottom rows), but during operation, the control signal 1210 and readout signal 1212 can address each element in the qubit assembly 1252.

[0121] The control and measurement system 1204 is an example of a classical computer system that can be used to perform various operations as described above, as well as other classical subroutines or computations, on the qubit component 1252. The control and measurement system 1204 includes one or more classical processors (e.g., classical processor 1214), one or more memories (e.g., memory 1216), and one or more I / O units (e.g., I / O unit 1218) connected via one or more data buses. The control and measurement system 1204 can be programmed to send a sequence of control signals 1210 to the qubit component, for example, to perform a selected series of quantum gate operations, and to receive a sequence of readout signals 1212 from the qubit component, for example, as part of performing a measurement operation.

[0122] Processor 1214 is configured to process instructions for execution within control and measurement system 1204. In some embodiments, processor 1214 is a single-threaded processor. In other embodiments, processor 1214 is a multi-threaded processor. Processor 1214 is capable of processing instructions stored in memory 1216.

[0123] Memory 1216 stores information within control and measurement system 1204. In some embodiments, memory 1216 includes computer-readable media, volatile memory cells, and / or non-volatile memory cells. In some cases, memory 1216 may include storage devices capable of providing mass storage for system 1204, such as hard disk drives, optical disk drives, storage devices shared by multiple computing devices (e.g., cloud storage devices) via a network, and / or other mass storage devices.

[0124] Input / output device 1218 provides input / output operations for control and measurement system 1204. Input / output device 1218 may include a D / A converter, an A / D converter, and an RF / microwave / optical signal generator, transmitter, and receiver, thereby sending control signals 1210 to and receiving readout signals 1212 from the qubit component, as is suitable for a physical scheme of a quantum computer. In some embodiments, input / output device 1218 may also include one or more network interface devices, such as Ethernet cards, serial communication devices (e.g., RS-232 ports), and / or wireless interface devices (e.g., 802.11 cards). In some embodiments, input / output device 1218 may include driver devices configured to receive input data and send output data to other external devices, such as keyboards, printers, and display devices.

[0125] Despite Figure 12An example control and measurement system 1204 has been described in this specification, but the implementation of the subject matter and functional operation described herein can be implemented in other types of digital electronic circuits, or in computer software, firmware, or hardware, including the structures disclosed herein and their structural equivalents, or combinations thereof.

[0126] Example system 1200 includes example classic processor 150. According to some embodiments, classic processor 1250 can be used to perform classic computational operations described herein, such as classic machine learning methods described herein.

[0127] The embodiments of the subjects and operations described in this specification can be implemented in digital electronic circuit systems, analog electronic circuit systems, suitable quantum circuit systems, or more generally quantum computing systems, in tangibly embodied software or firmware, in computer hardware (including the structures disclosed in this specification and their equivalents), or in combinations of one or more of them. The term "quantum computing system" can include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0128] The embodiments of the subject matter described in this specification can be implemented as one or more computer programs, that is, one or more modules of computer program instructions encoded on a tangible, non-transitory storage medium, for execution by a data processing device or for controlling the operation of a data processing device. The computer storage medium may 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 thereof. Alternatively or additionally, the program instructions may be encoded on artificially generated propagating signals capable of encoding digital and / or quantum information, such as machine-generated electrical, optical, or electromagnetic signals, which are generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing device.

[0129] The terms quantum information and quantum data refer to information or data carried, held, or stored in quantum systems, the smallest nontrivial system being a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be appropriately approximated as a two-level system in the appropriate context. Such quantum systems can include multi-level systems, for example, systems with two or more levels. As examples, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational fundamental state is identified using the ground state and a first excited state; however, it should be understood that other settings where the computational state is identified using a higher-level excited state are possible.

[0130] The term "data processing device" refers to digital and / or quantum data processing hardware, and includes all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, such as programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or further include special-purpose logic circuitry, such as FPGAs (Field-Programmable Gate Arrays), ASICs (Application-Specific Integrated Circuits), or quantum simulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special-purpose quantum computer that does not have the capability to perform general-purpose quantum computing. In addition to hardware, the device may optionally include code that creates 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 thereof.

[0131] A digital computer program, which may also be referred to or described as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for a digital computing environment. A quantum computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language such as QCL or Quipper.

[0132] Computer programs can, but do not necessarily, correspond to files in a file system. A program can be stored as a portion of a file containing 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 coordinating files, such as a file storing one or more modules, subroutines, or code sections. Computer programs can be deployed to execute on a single computer or on multiple computers located at one site or distributed across multiple sites and interconnected via digital and / or quantum data communication networks. A quantum data communication network is understood as a network that can transmit quantum data using quantum systems (e.g., qubits). Typically, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum and digital data.

[0133] The processes and logic flows described in this specification can be executed by one or more programmable computers, operating in conjunction with one or more processors where appropriate, to execute one or more computer programs to perform functions by manipulating input data and generating outputs. The processes and logic flows can also be executed by dedicated logic circuitry (e.g., FPGA or ASIC) or a quantum simulator, or by a combination of dedicated logic circuitry or a quantum simulator and one or more programmable digital and / or quantum computers, and the apparatus can also be implemented as dedicated logic circuitry or a quantum simulator.

[0134] For a system of one or more computers “configured” to perform a specific operation or action, it means that the system has software, firmware, hardware, or a combination thereof installed thereon, which, in operation, cause the system to perform the operation or action. For one or more computer programs configured to perform a specific operation or action, it means that the one or more programs include instructions that, when executed by a data processing device, cause the device to perform the operation or action. For example, a quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform an operation or action.

[0135] A computer suitable for executing computer programs can be based on a general-purpose or special-purpose processor or any other type of central processing unit. Typically, the central processing unit receives instructions and data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (such as photons), or a combination thereof.

[0136] A computer's components include a central processing unit (CPU) for executing or running instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The CPU and memory may be supplemented or incorporated into dedicated logic circuitry or a quantum simulator. Typically, a computer will also include one or more high-capacity storage devices (e.g., magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information) for storing data, or operatively coupled to receive data from or transfer data to, or both. However, a computer does not necessarily need to have such devices.

[0137] Quantum circuit elements (also known as quantum computing circuit elements) include circuit elements used to perform quantum processing operations. That is, quantum circuit elements are configured to utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on data in a nondeterministic manner. Some quantum circuit elements (such as qubits) can be configured to simultaneously represent and manipulate information from more than one state. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUIDs or DC-SQUIDs).

[0138] In contrast, classical circuit elements typically 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, logic, and / or input / output operations on data, where 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 via electrical or electromagnetic connections. Examples of classical circuit elements include CMOS-based circuit elements, fast single-throughput quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and ERSFQ devices, which are energy-efficient versions of RSFQs that do not use bias resistors.

[0139] In some cases, some or all of quantum and / or classical circuit elements can be realized using, for example, superconducting quantum and / or classical circuit elements. Fabrication of superconducting circuit elements may require the deposition of one or more materials, such as superconductors, dielectrics, and / or metals. Depending on the materials chosen, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxy, as well as other deposition processes. Processes used to fabricate the circuit elements described herein may require the removal of one or more materials from the device during fabrication. Depending on the material to be removed, the removal process may include, for example, wet etching, dry etching, or stripping processes. The materials forming the circuit elements described herein can be patterned using known photolithography techniques (e.g., photolithography or electron beam lithography).

[0140] During the operation of a quantum computing system using superconducting quantum circuit elements and / or superconducting classical circuit elements (such as those described herein), the superconducting circuit elements are cooled within a cryostat to a temperature that allows the superconducting material to exhibit superconducting properties. A superconducting (or alternatively superconducting) material can be understood as a material that exhibits superconducting properties at or below its superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature of 1.2 Kelvin) and niobium (superconducting critical temperature of 9.3 Kelvin). Therefore, superconducting structures (such as superconducting traces and superconducting ground planes) are formed from materials that exhibit superconducting properties at or below their superconducting critical temperature.

[0141] In some implementations, classical circuit elements electrically and / or electromagnetically coupled to the quantum circuit elements can be used to provide control signals for the quantum circuit elements (e.g., qubits and qubit couplers). The control signals can be provided in digital and / or analog form.

[0142] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memories, media, and memory devices, including, for 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 and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data with high fidelity and efficiency for extended periods, such as a light-matter interface where light is used for transmission and matter is used for storage and preservation of quantum characteristics of the quantum data, such as superposition or quantum coherence.

[0143] Control of the various systems or portions thereof described in this specification can be implemented in a computer program product comprising instructions stored on one or more non-transitory machine-readable storage media and executable on one or more processing devices. The systems or portions thereof described in this specification can each be implemented as an apparatus, method, or system, which may include one or more processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0144] While this specification contains numerous details of specific embodiments, these should not be construed as limiting the scope that can be claimed, but rather as descriptions of features that may be specific to particular embodiments. Some features described in this specification in the context of individual embodiments may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented individually or in any suitable sub-combination in multiple embodiments. Furthermore, although features may be described above as functioning in certain combinations and even initially claimed in this way, in some cases one or more features from the claimed combination may be removed from the combination, and the claimed combination may be for sub-combinations or variations thereof.

[0145] Similarly, although the operations are depicted in a specific order in the accompanying drawings, this should not be construed as requiring these operations to be performed in the specific order shown or sequentially, or to perform all the shown operations to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of the various system modules and components in the above embodiments should not be construed as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0146] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the appended claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired result. As an example, the processes depicted in the drawings do not necessarily require the specific order or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for training a quantum generative adversarial network to learn a target quantum state, the method being performed by one or more quantum computers comprising multiple qubits, the method comprising: The parameters of the quantum generative adversarial network are iteratively adjusted until the value of the quantum generative adversarial network loss function converges. The quantum generative adversarial network includes: a generator network, comprising quantum hardware of one or more quantum computers configured to apply a first parameterized quantum circuit to an initial quantum state to output a first quantum state approximating a target quantum state; and a discriminator network, comprising quantum hardware of one or more quantum computers configured to apply a second parameterized quantum circuit to the discriminator network. Each iteration includes: An entanglement operation is performed on the discriminator network input of a discriminator network in a quantum generative adversarial network to measure the fidelity of the discriminator network input, wherein the discriminator network input includes a target quantum state and a first quantum state output from the generator network in the quantum generative adversarial network; and Mini-maximum optimization of the quantum generative adversarial network loss function is performed to update the parameters of the quantum generative adversarial network, where the quantum generative adversarial network loss function depends on the fidelity of the measurement of the discriminator network input.

2. The method according to claim 1, wherein, Each iteration also includes: An initial quantum state is processed by a generator network to output a first quantum state, the processing including applying a first quantum circuit to the initial quantum state, wherein i) the first quantum circuit is a parameterized quantum circuit, and the parameters of the first quantum circuit constitute the parameters of the generator network included in the parameters of the quantum generative adversarial network.

3. The method according to claim 1, wherein, Entanglement operations include unassisted swap tests.

4. The method according to claim 1, wherein, The loss function of a quantum generative adversarial network consists of 1 minus the fidelity of the measurement input to the discriminator network.

5. The method according to claim 1, wherein, Mini-maximum optimization of the loss function in quantum generative adversarial networks includes: The generator network parameters are fixed to values ​​determined in previous iterations, and the quantum generative adversarial network loss function is maximized relative to the discriminator network parameters to determine updated values ​​for the discriminator network parameters for this iteration; and The discriminator network parameters are fixed to the updated values ​​for the discriminator network parameters for this iteration, and the quantum generative adversarial network loss function is minimized relative to the generator network parameters to determine the updated values ​​for the generator network parameters for this iteration.

6. The method according to claim 1, wherein, Mini-maximum optimization of the loss function in quantum generative adversarial networks includes: The discriminator network parameters are fixed to the values ​​corresponding to the perfect swap test, and the quantum generative adversarial network loss function is minimized relative to the generator network parameters to determine the initial update values ​​of the generator network parameters for this iteration. The generator network parameters are fixed to their initial update values, and the quantum generative adversarial network loss function is maximized relative to the discriminator network parameters to determine the update values ​​of the discriminator network parameters for this iteration; and The discriminator network parameters are fixed to the updated values ​​for the discriminator network parameters for this iteration, and the quantum generative adversarial network loss function is minimized relative to the generator network parameters to determine the updated values ​​for the generator network parameters for this iteration.

7. The method according to claim 1, wherein, The target quantum state includes a superposition state, and the method further includes: generating the target quantum state by a generator network based on trained generator network parameters.

8. The method according to claim 7, further comprising: The generated target quantum state is used to train the quantum neural network.

9. The method according to claim 1, wherein, The parameters of the quantum generative adversarial network are iteratively adjusted until the value of the quantum generative adversarial network loss function converges to generate trained generator network and discriminator network parameters. The method also includes using the generator network and generating a target state based on the trained generator network parameters.

10. The method according to any one of claims 1 to 9, wherein, Performing minimax optimization of the loss function of a quantum generative adversarial network (GAN) to update the parameters of the GAN includes performing multiple circuit evaluations to compute the gradients of the GAN's parameters.

11. A system for training a quantum generative adversarial network to learn a target quantum state, comprising: Quantum computing devices, including qubit components; as well as Classical computer systems, including control and measurement systems, The system is configured to perform the method described in any of the preceding claims.

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