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Variational Quantum Circuits for Deep Reinforcement Learning

Samuel Yen-Chi Chen, Chao-Han Huck Yang, Jun Qi, Pin-Yu Chen, Xiaoli Ma, Hsi-Sheng Goan

arXiv:1907.00397v3cs.LGcs.AIquant-phstat.ML

TL;DR

Large quantum circuits are difficult to simulate and deploy on current platforms, motivating feasible quantum algorithms for noisy intermediate-scale devices. This paper adapts deep reinforcement learning to variational quantum circuits and demonstrates reduced parameter counts and memory consumption in the considered environments.

  • Problem

    Current quantum platforms cannot faithfully employ large circuits, while deep reinforcement-learning value-function training can be difficult to converge with nonlinear approximators.

  • Method

    The paper represents experience replay, target networks, and action-value approximation with variational quantum circuits using quantum encoding and iterative classical optimization.

  • Results

    The variational quantum deep Q-learning framework uses as few as O(n) parameters in the considered environments, versus O(n^3) for tabular Q-learning and O(n^2) for neural-network deep Q-learning.

  • Takeaways & Limitations

    The study provides a proof-of-principle demonstration that variational quantum circuits can approximate deep Q-values for reinforcement learning on near-term quantum platforms.

  • Takeaways & Limitations

    The experiments use computational-basis encoding, which does not fully exploit quantum advantages and leaves amplitude encoding as future work.

Abstract

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The state-of-the-art machine learning approaches are based on classical von Neumann computing architectures and have been widely used in many industrial and academic domains. With the recent development of quantum computing, researchers and tech-giants have attempted new quantum circuits for machine learning tasks. However, the existing quantum computing platforms are hard to simulate classical deep learning models or problems because of the intractability of deep quantum circuits. Thus, it is necessary to design feasible quantum algorithms for quantum machine learning for noisy intermediate scale quantum (NISQ) devices. This work explores variational quantum circuits for deep reinforcement learning. Specifically, we reshape classical deep reinforcement learning algorithms like experience replay and target network into a representation of variational quantum circuits. Moreover, we use a quantum information encoding scheme to reduce the number of model parameters compared to classical neural networks. To the best of our knowledge, this work is the first proof-of-principle demonstration of variational quantum circuits to approximate the deep $Q$-value function for decision-making and policy-selection reinforcement learning with experience replay and target network. Besides, our variational quantum circuits can be deployed in many near-term NISQ machines.

I. INTRODUCTION · II. REINFORCEMENT LEARNING

The paper motivates variational quantum circuits for deep reinforcement learning on noisy intermediate-scale quantum machines. It introduces reinforcement learning as sequential agent–environment interaction in which policies select actions to maximize expected discounted returns.

  • I. INTRODUCTION: Quantum computing platforms face difficulties faithfully executing large-scale quantum circuits because of limited quantum error correction.This motivates methods compatible with available quantum platforms.
  • I. INTRODUCTION: Variational quantum circuits use iterative optimization to reduce circuit depth and enable hybrid quantum-classical algorithms on NISQ machines.Prior work also demonstrated classification and clustering with significantly fewer parameters on NISQ systems.
  • I. INTRODUCTION: The work focuses on empowering deep reinforcement learning on NISQ computation for automatic policy learning under uncertainty.The proposed approach uses variational quantum circuits for complex sequential decision-making systems.
  • II. REINFORCEMENT LEARNING: Reinforcement learning models an agent interacting with an environment over discrete time steps, receiving states and selecting actions according to a policy.A policy maps a state s_t to an action a_t and may define a probability distribution over actions.
  • II. REINFORCEMENT LEARNING: After each action, the agent receives a next state and scalar reward, continuing until a terminal state; an episode spans this process from an initial state.Episodes begin from randomly selected initial states and proceed through terminal states.
  • II. REINFORCEMENT LEARNING: The total discounted return from time t is formed by weighting future rewards by γ, where γ lies in (0, 1] and controls their influence.Large γ values retain future rewards, whereas small γ values allow them to be ignored after a few steps.
  • II. REINFORCEMENT LEARNING: The Q-value function Q^π(s, a) is the expected return for selecting action a in state s under policy π, while the optimal value maximizes over policies.The agent’s training objective is to maximize expected return from each state.

A. Q-LEARNING · B. STATE-ACTION-REWARD-STATE-ACTION (SARSA) · C. DEEP Q-LEARNING

The section introduces Q-learning and SARSA as temporal-difference reinforcement-learning methods, then explains why deep Q-learning uses neural-network function approximation and stabilization mechanisms for large state or action spaces.

  • A. Q-LEARNING: Q-learning is a model-free, off-policy algorithm that initializes Q arbitrarily and updates action values using observed rewards, transitions, and learning rate α.Actions can be selected using an ϵ-greedy policy derived from Q.
  • B. STATE-ACTION-REWARD-STATE-ACTION (SARSA): SARSA updates the policy based on undertaken actions, transitioning from (s_t, a_t) to (s_t+1, a_t+1) with temporal difference and learning rate α.Its Q value combines the next-step reward with discounted future reward from the next state-action observation.
  • C. DEEP Q-LEARNING: A tabular action-value function contains s × a entries, but this representation becomes unfeasible when state or action spaces are large or continuous.Function approximators such as neural networks are used in those settings.
  • C. DEEP Q-LEARNING: Deep Q-learning parameterizes Q(s, a; θ) with neural networks, whose parameters θ are obtained through iterative optimization methods.Neural-network function approximation has been applied to tasks including video-game playing.
  • C. DEEP Q-LEARNING: Deep reinforcement learning predicts Q(s_t, a_t; θ) against target r_t + γ max_a′ Q(s_t+1, a′; θ−), but nonlinear approximation can make the loss hard to converge or divergent.θ denotes the policy-network parameters, while θ− denotes target-network parameters.
  • C. DEEP Q-LEARNING: The target network updates θ− only at finite time steps, helping stabilize Q-value training because its target is relatively stationary.This mechanism is presented alongside experience replay as addressing deep Q-learning issues.
  • C. DEEP Q-LEARNING: Experience replay stores transitions (s_t, a_t, r_t, s_t+1), randomly samples batches, and updates Q-learning parameters using gradient descent on an MSE loss.The loss uses the target-network estimate and the sampled batch of experiences.

III. TESTING ENVIRONMENTS · A. FROZEN LAKE · B. COGNITIVE RADIO

The study evaluates its reinforcement-learning agent in two environments: the simple Frozen Lake maze and the real-world cognitive-radio channel-selection task. These environments define distinct observations, actions, rewards, and episode-ending conditions for testing.

  • III. TESTING ENVIRONMENTS: The evaluation uses Frozen Lake and cognitive radio because Frozen Lake is simple, commonly tested, and feasible to simulate on available quantum simulators and NISQ devices.The environments are selected to study reinforcement-learning agent performance under feasible testing conditions.
  • A. FROZEN LAKE: Frozen Lake is an OpenAI Gym maze where the agent must travel from start S to goal G while avoiding holes H.Entering a hole gives a large negative reward and terminates the episode.
  • A. FROZEN LAKE: Frozen Lake offers four actions: LEFT, DOWN, RIGHT, and UP.The variational quantum circuit’s action-selection procedure is described later in Sec. VI-A.
  • A. FROZEN LAKE: +1.0 reward is given for reaching the Frozen Lake goal, −0.2 for entering a hole, and −0.01 for each step.The step penalty encourages the agent to find the shortest path; the environment is non-slippery.
  • B. COGNITIVE RADIO: In cognitive radio, the agent selects a channel that is not occupied or interfered with by a primary user.A successful selection earns +1, an unsuccessful selection earns −1, and three failed selections terminate the episode.
  • B. COGNITIVE RADIO: Cognitive-radio observations use ns3 statistics and channel capacity for a customized channel count n.For n = 4, the state [1 0 0 0] indicates a primary user on the first channel.
  • B. COGNITIVE RADIO: The cognitive-radio action is selecting one channel for the secondary user from n available channels.The corresponding variational quantum circuit action-selection method is described in Sec. VI-A.
  • B. COGNITIVE RADIO: The testing environments use −1 for a cognitive-radio collision and +1 for no collision, with a maximum achievable score of 100.The reward settings are summarized in Table 1 alongside the Frozen Lake rewards.

C. VARIATIONAL QUANTUM DEEP Q-LEARNING · IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING

The paper presents a hybrid variational quantum-classical deep reinforcement learning framework in which quantum circuits support action selection while classical computation optimizes circuit parameters. It motivates this approach for NISQ devices by addressing quantum-circuit simulation limits, resource demands, and practical DRL mechanisms such as target networks and experience replay.

  • C. VARIATIONAL QUANTUM DEEP Q-LEARNING: The proposed DRL agent combines quantum and classical components and is evaluated on frozen-lake and cognitive-radio environments as a proof-of-principle study.The framework is described as general and potentially applicable to more complicated tasks.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: Variational quantum circuits use tunable parameters optimized iteratively by a classical computer, analogous to weights in artificial neural networks.Their circuit depth is flexible, supporting hybrid quantum-classical computation.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: Variational-circuit quantum machine learning can circumvent complex quantum errors in available NISQ devices and model function approximators, classifiers, and some quantum-many-body systems.The cited prior results motivate variational circuits as a practical approach despite limited error correction and fault tolerance.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: A 100-qubit quantum circuit has a computational state space of dimensions 2^100, making large quantum circuits difficult to simulate on classical computers.The passage contrasts this limitation with Google’s demonstration involving a 53-qubit quantum computer.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: The quantum circuit takes two parameter sets and produces measurement results that determine possible actions, while a classical computer performs optimization.This division of responsibilities forms the main proposed VQ-DQL architecture.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: The main circuit parameters are updated every step, whereas target circuit parameters use the same architecture and are updated per 20 steps.The framework also incorporates experience replay through replay memory.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: The generic VQ-DQN architecture uses Rx and Rz rotations, CNOT entanglement gates, and a general single-qubit unitary gate with three parameters.The figure also distinguishes state-preparation parameters from iteratively optimized parameters.
  • IV. VARIATIONAL QUANTUM CIRCUITS AND DEEP Q-LEARNING: Artificial neural networks are universal approximators, but their hidden layers may require many neurons and therefore many parameters.The paper emphasizes that model resource consumption matters in addition to approximation capability.

A. COMPUTATIONAL BASIS ENCODING AND QUANTUM CIRCUIT FOR THE FROZEN LAKE PROBLEM

The Frozen Lake variational circuit encodes each discrete environment state into a computational-basis quantum state, then uses parameterized gates and entanglement to represent action values. Its design balances expressive circuit depth against low depth and parameter count for NISQ implementation.

  • Computational basis encoding: The 16 Frozen Lake states are labeled 0–15, converted to binary, and encoded using a 4-qubit computational-basis register.For example, state 11 becomes binary 1011 and is encoded as |1⟩⊗|0⟩⊗|1⟩⊗|1⟩.
  • Variational quantum deep Q learning: The quantum deep Q-learning procedure initializes replay memory and a parameterized action-value circuit, encodes states, stores transitions, samples minibatches, and performs gradient descent.Actions use ε-greedy selection, and non-terminal targets include γ maxa′ Q(sj+1, a′; θ).
  • Quantum circuit architecture: CNOT gates entangle qubits, while the circuit’s αi, βi, and γi parameters are optimized iteratively.The general single-qubit unitary gate contains these three optimization parameters.
  • Circuit depth and expressivity: The variational circuit can vary in depth: shallow circuits may represent a task’s solution space, whereas deeper circuits may improve performance.Choosing an effective circuit that maintains low depth and few parameters remains challenging for general tasks.

B. QUANTUM CIRCUIT FOR COGNITIVE RADIO NETWORKS

The cognitive-radio VQ-DQN encodes each observation as a channel–time pair, yielding n^2 possible states under a periodic channel-changing pattern. Its circuits use channel-dependent qubit counts and two layers, while actions are selected from expectation values of n qubits.

  • State and observation encoding: Each observation combines the selected channel with temporal information, represented as (channel, time).The channel-changing pattern is assumed periodic, with n time steps per full cycle.
  • Circuit architecture: The cognitive-radio variational circuits use different qubit counts to fit the number of channels and two layers for every channel configuration.The architecture otherwise matches the frozen-lake circuit.
  • State and observation encoding: n^2 possible states arise because the agent observes n channels across n time steps in the periodic cycle.
  • Action selection: The agent has n possible channel-selection actions, whose VQ-DQN values are determined from expectation values of n qubits.For n = 3, repeated measurements of only the first three qubits estimate the expectation values because nq > n.

VI. EXPERIMENTS AND RESULTS

The experiments evaluate variational quantum circuits for DQL on the frozen-lake environment across three configurations, with the proposed agent reaching an optimal policy in one configuration.

  • Frozen-lake experiment: Figure 7 evaluates the variational quantum circuits-based DQL agent on the frozen-lake experiment across environment configurations (a), (b), and (c).The configurations correspond to those shown in Fig. 1.
  • Frozen-lake experiment: 0.9 total reward: the proposed agent reaches the optimal policy at the 198th iteration in configuration (a).This result is shown in the left panel of subfigure (a).

A. ENVIRONMENT SETUP AND ACTION SELECTION

The experiments use frozen-lake and periodic-interference cognitive-radio environments configured according to the variational circuit architectures. The agent selects the action corresponding to the qubit with the largest expectation value, using ensemble measurements rather than single-shot outcomes.

  • Environment setup: Experiments cover frozen-lake testing and cognitive-radio environments with periodic interference sweeping channels from 1 to n.The circuit architecture follows the environment and channel count, including the n = 4 cognitive-radio case.
  • Action selection: The agent chooses the action indexed by the measurement output wire with the largest expectation value.For an n-qubit circuit, measurement outputs form an n-tuple, with output wires indexed from 0 to n − 1.
  • Action selection: Expectation values determine actions instead of stochastic 0-or-1 outcomes from individual qubit measurements.Single-shot outcomes are insufficient for deciding the next move because they are truly stochastic.
  • Expectation estimation: Quantum simulators can calculate expectation values analytically and deterministically, whereas real quantum computers estimate them from many measurements.The passage names PennyLane and IBM Qiskit as example simulators.
  • Expectation estimation: Determining each qubit’s expectation value avoids learning the full n-qubit probability distribution, whose measurement requirement can scale exponentially with n.The algorithm needs per-qubit expectation values rather than all state probabilities.

B. NUMERICAL SIMULATION

The numerical simulations use PennyLane, PyTorch, and OpenAI Gym to evaluate the variational quantum circuits in frozen-lake and cognitive-radio environments. Training uses RMSprop with specified optimization settings, experience-replay batch size 5, and epsilon-greedy exploration that shifts from exploration toward exploitation.

  • Simulation setup: The circuits are numerically simulated with PennyLane, assisted by PyTorch for linear-algebra manipulation, and tested in OpenAI Gym environments.The experiments use the low-complexity frozen-lake environment for proof-of-concept experiments and the cognitive-radio environment for further study.
  • Optimization settings: RMSprop uses learning rate = 0.01, alpha = 0.99, and eps = 10−8 for gradient-descent optimization.The experience-replay batch-size is 5.
  • Exploration strategy: The frozen-lake environment uses an ϵ-greedy strategy, while cognitive-radio updates ϵ at every step.The supplied passages state the update frequency but do not include the corresponding equations.
  • Exploration strategy: Initial ϵ = 1.0 encourages exploration in early episodes before shifting toward exploitation in later episodes.This initialization is described as supporting the exploration-to-exploitation transition.

C. SIMULATION WITH NOISE · D. PERFORMANCE ANALYSIS · E. RUNNING ON A QUANTUM COMPUTER

The study evaluates variational quantum DRL under realistic noise, analyzes training performance across frozen-lake and cognitive-radio tasks, and tests trained models on IBM quantum hardware. Results indicate convergence, training stability, and effective real-device execution despite noise and simulation-based training.

  • C. SIMULATION WITH NOISE: The authors simulate variational quantum circuits with IBM quantum-computer noise models using Qiskit Aer to investigate robustness on near-term NISQ devices.The noisy backend replaces the noiseless simulation backend while retaining the previous experiment setting.
  • C. SIMULATION WITH NOISE: Noise deviations can be absorbed during classical iterative optimization, while action selection depends only on the qubit with the largest expectation value.This avoids requiring exact expectation values for every qubit.
  • D. PERFORMANCE ANALYSIS: In frozen-lake configuration (a), the agent converges to total reward 0.9 after the 198th episode during a 500-episode experiment.The other two configurations show a similar convergence pattern, although several sub-optimal results occur.
  • D. PERFORMANCE ANALYSIS: Training stability is assessed using temporary mean rewards and standard-deviation boundaries computed from the last 100 episodes.When fewer than 100 episodes are available, all episodes are included; standard deviation and mean are shown in gray and blue, respectively.
  • D. PERFORMANCE ANALYSIS: In cognitive-radio experiments with 2, 3, 4, and 5 possible channels, agents converge to the optimal total reward at around 100 iterations or episodes.Each situation is evaluated over 500 episodes, with additional testing on two other 4-channel environment configurations.
  • E. RUNNING ON A QUANTUM COMPUTER: On IBM Q ibmq-valencia, the trained models perform well despite being trained without quantum noise in simulation for both cognitive-radio and frozen-lake experiments.The cognitive-radio test uses configuration (c) in the 4-channel case, while the frozen-lake test uses configuration (c).
  • E. RUNNING ON A QUANTUM COMPUTER: The trained variational quantum DRL models are uploaded to IBM Q to test feasibility on a real quantum computer, with five episodes run for the 4-channel cognitive-radio configuration.The cloud platform’s limited resources restrict the number of episodes in this experiment.
  • E. RUNNING ON A QUANTUM COMPUTER: The real-device cognitive-radio execution achieves the same total reward as PennyLane or Qiskit simulation, supporting robustness to hardware noise.The algorithm compares qubits by their largest expectation value rather than requiring exact expectation values.

F. QUANTUM ADVANTAGE ON MEMORY CONSUMPTION · VII. DISCUSSIONS · A. OVERVIEW OF QUANTUM MACHINE LEARNING

The proposed variational quantum deep Q-learning circuits target lower memory consumption than classical reinforcement-learning models through data encoding schemes, while remaining suitable for near-term NISQ constraints. The work also distinguishes this approach from earlier quantum reinforcement-learning schemes that are difficult to implement on NISQ machines.

  • VII. DISCUSSIONS: Current NISQ machines constrain the proposed circuit because missing quantum error correction makes deeper quantum-gate architectures more error-prone.The passage therefore motivates circuits that are not deep in quantum gates and cannot have many parameters compared with classical neural networks.
  • F. QUANTUM ADVANTAGE ON MEMORY CONSUMPTION: 56.25% fewer parameters are required by VQ-DQL than tabular Q-learning in the frozen-lake environment.VQ-DQL uses 28 parameters versus a 16 × 4 = 64-entry tabular Q-learning table.
  • F. QUANTUM ADVANTAGE ON MEMORY CONSUMPTION: VQ-DQN with computational basis encoding uses a parameter count that grows linearly with the input dimension n.This scaling is reported for the tested cognitive-radio cases with n ≤5 channels.
  • F. QUANTUM ADVANTAGE ON MEMORY CONSUMPTION: VQ-DQN with amplitude encoding can use logarithmically fewer parameters than classical models.The passage describes this as harvesting full logarithmic parameter savings compared with classical models.
  • F. QUANTUM ADVANTAGE ON MEMORY CONSUMPTION: In the cognitive-radio experiment, the proposed quantum circuit uses fewer parameters than the selected classical counterpart.The classical architecture is selected from the neural-network model used in the cited ns3 simulation framework.
  • VII. DISCUSSIONS: The quantum advantage is defined as lower memory consumption, corresponding to fewer parameters in the quantum circuits.Its parameter advantage depends on the selected data encoding scheme.
  • A. OVERVIEW OF QUANTUM MACHINE LEARNING: Earlier quantum reinforcement-learning schemes use amplitude amplification for action selection but are hard to implement on NISQ machines.Those schemes also require encoding the environment into a quantum superposition state, which is not always possible during classical agent interaction.

B. MORE COMPLEX TESTING ENVIRONMENTS … VIII. CONCLUSIONS

The paper evaluates VQ-DQN in simple maze, frozen-lake, and cognitive-radio environments, while discussing scaling, classical simulability, amplitude encoding, and concluding that the framework reduces memory and model parameters. It presents a first demonstration of variational quantum circuits approximating deep Q-values with experience replay and a target network.

  • B. MORE COMPLEX TESTING ENVIRONMENTS: VQ-DQN is applied to a simple maze, OpenAI Gym’s frozen-lake environment, and a cognitive-radio spectrum-control problem using ns3-gym.Frozen Lake is selected for low computational complexity in proof-of-principle quantum DQL experiments.
  • B. MORE COMPLEX TESTING ENVIRONMENTS: The study uses simple testing environments because complex reinforcement-learning benchmarks require intractably many qubits for classical simulation and exceed current commercial quantum devices.The authors suggest investigating such environments when larger quantum machines become available.
  • C. SCALING UP THE ARCHITECTURE: Inference requires measuring each qubit’s expectation value to select the next action, with real quantum hardware requiring repeated samplings rather than analytical classical simulation.For an n-qubit circuit with n output actions, the passage gives 1024 as a possible fixed sampling count.
  • D. SIMULABILITY ON A CLASSICAL COMPUTER: With a fixed number of variational layers, the proposed circuit’s CNOT-gate count scales linearly with n, making it non-constant-depth.The circuit also contains gates beyond the Clifford group and therefore is not obviously classical.
  • E. FUTURE WORK: AMPLITUDE ENCODING SCHEME: Computational-basis encoding does not fully exploit quantum advantages, although constrained quantum-simulator experiments support the feasibility of quantum circuits for deep reinforcement learning.The passage reports empirical quantum advantages over tabular Q-learning, but the supplied text truncates the comparison.
  • E. FUTURE WORK: AMPLITUDE ENCODING SCHEME: Amplitude encoding could reduce circuit parameters to as small as poly(log n), compared with poly(n) parameters in a standard neural network.Future work includes applying amplitude encoding to more complex input data and variational circuits to more sophisticated problems.
  • VIII. CONCLUSIONS: The work presents the first demonstration of variational quantum circuits approximating the deep Q-value function with experience replay and a target network.For the considered environments, the variational quantum deep Q-learning model uses parameters as small as O(n), according to the supplied conclusion.

APPENDIX A NOISE INFORMATION OF THE IBM Q MACHINES USED IN THE EXPERIMENTS · A. DEVICE PROPERTIES FOR THE NOISY SIMULATION · B. DEVICE PROPERTIES FOR THE QUANTUM INFERENCE

The appendix documents IBM Q device noise data used for noisy simulation and quantum inference, including qubit coherence, frequencies, gate errors, gate lengths, readout errors, and CNOT properties. It also specifies the machines, experiment contexts, download dates, and single-qubit gate definitions relevant to these evaluations.

  • APPENDIX A NOISE INFORMATION OF THE IBM Q MACHINES USED IN THE EXPERIMENTS: Noisy simulation used IBM Qiskit noise data from the 20-qubit ibmq-poughkeepsie machine, downloaded on 2019-07-28.The data supported numerical simulation in Sec. VI-C.
  • A. DEVICE PROPERTIES FOR THE NOISY SIMULATION: The noisy-simulation noise model includes T1, T2, qubit frequency, single-qubit gate error, gate length, readout error, coupling, and CNOT data.These properties are distributed across Tables 6, 8, and 9.
  • A. DEVICE PROPERTIES FOR THE NOISY SIMULATION: Single-qubit gates are represented by the general unitary U(θ, φ, λ), with U1, U2, and U3 instantiated from u1, u2, and u3.The appendix defines u1(λ) = U(0, 0, λ), u2(φ, λ) = U(π/2, φ, λ), and u3(θ, φ, λ) = U(π/2, φ, λ).
  • A. DEVICE PROPERTIES FOR THE NOISY SIMULATION: The listed noisy-simulation gates are U1 = u1(π/2), U2 = u2(π/2, π/2), and U3 = u3(π/2, π/2, π/2).U1 is implemented as a software frame change, so its gate time and gate error are both zero.
  • B. DEVICE PROPERTIES FOR THE QUANTUM INFERENCE: Quantum inference experiments used the real 5-qubit ibmq-valencia machine, with coherence, frequency, single-qubit, readout, coupling, and CNOT properties recorded in Tables 7, 10, and 11.These data correspond to the experiments in Sec. VI-E.
  • A. DEVICE PROPERTIES FOR THE NOISY SIMULATION: The appendix records two-qubit coupling and CNOT error and gate-length data for ibmq-poughkeepsie, noting that values vary across coupling pairs.The variation is attributed to different coupling strengths and qubit frequencies in the pairs.
  • A. DEVICE PROPERTIES FOR THE NOISY SIMULATION: For noisy simulation, Id, U1, U2, and U3 gate lengths are 113.7778 ns, 0.0 ns, 103.1111 ns, and 206.2222 ns, respectively.The same single-qubit gate has the same length for every qubit, while different gates have different lengths.
  • B. DEVICE PROPERTIES FOR THE QUANTUM INFERENCE: For quantum inference, the listed Id, U1, U2, and U3 gate lengths are 35.55556 ns, 0.0 ns, 35.55556 ns, and 71.11111 ns, respectively.The table identifies these as single-qubit gate lengths for ibmq-valencia.
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