Differentiable Quantum Architecture Search

• Fixed variational ansätze optimize parameters without adapting circuit structure, despite differences in expressive power and entangling capability.Source: §1 · p. 1
• NISQ circuit design must account for problem structure, decoherence, limited qubit connectivity, and resource costs such as two-qubit gates and circuit depth.Source: §1 · p. 1
• QAS denotes automated parameterized-circuit design, while quantum architecture search covers broader quantum-circuit design beyond variational ansätze.Source: §1 · p. 1
• Existing QAS approaches include greedy, evolutionary or genetic, and reinforcement-learning methods, paralleling common neural architecture search strategies.Source: §1 · p. 1
• DQAS relaxes quantum architecture choices into a differentiable domain, enabling a general, fully automated, end-to-end circuit-design workflow.Source: §1 · pp. 1–2

Differentiable Neural Architecture Search

• NAS seeks task-specific high-performing neural architectures using reinforcement learning, evolutionary algorithms, or differentiable relaxation.Source: §2 · p. 2
• DARTS relaxes discrete architecture components into continuous variables optimized by differentiation and gradient descent, reducing training time and hardware requirements.Source: §2 · p. 2

Related works on QAS

• Earlier QAS studies mainly used evolutionary, greedy, or reinforcement-learning methods for QAOA and VQE-related circuit-structure tasks.Source: §2 · pp. 2–3
• These QAS methods search discrete quantum architectures, motivating DQAS as an end-to-end differentiable alternative.Source: §2 · p. 3

Brief review on QAOA

• QAOA alternates phase or cost evolution with mixer evolution, using variational parameters to address combinatorial optimization problems.Source: §2 · p. 3
• Generalized QAOA separates preparation, mixer, phase/problem, and cost Hamiltonians, creating layouts that DQAS can automatically search beyond vanilla QAOA.Source: §2 · p. 3
• QAOA uses relatively few variational parameters, potentially easing training compared with same-depth hardware-efficient VQE or quantum neural-network ansätze.Source: §2 · p. 3

Overview

• DQAS treats circuit design as bi-optimization: it selects unitaries for a circuit layout while identifying suitable variational parameters.Source: §3 · p. 4

Circuit encoding and operation pool

• The framework represents circuits as sequences of unitary placeholders filled from an operation pool, including one-, two-, or higher-level parameterized blocks.Source: §3 · p. 4
• Each placeholder can reuse an operation-pool unitary, allowing DQAS to assemble circuits with replacement.Source: §3 · p. 4

Objectives

• Objectives cover hybrid quantum-classical tasks, supervised learning, state preparation, and circuit compilation, including expectation-based losses and reference-circuit fidelity.Source: §3 · pp. 4–5

Sampling the structures

• DQAS samples discrete structures k from a probabilistic model parameterized by continuous α, then optimizes α and circuit parameters θ through automatic differentiation.Source: §3 · p. 5

Filling the circuit parameters

• Parameter sharing stores operation parameters in a shared pool, while Monte Carlo architecture sampling reaches an exponential architecture space with moderate resources.Source: §3 · pp. 5–6

Probabilistic models

• The default independent-category probabilistic model omits explicit inter-layer correlations; richer energy-based or autoregressive models are left for future work.Source: §3 · p. 6

Applications

• DQAS is presented as a versatile tool for quantum circuit design, with examples spanning unitary decomposition, quantum error mitigation, and hybrid quantum-classical algorithms.The implementations use quantum simulation backends based on Cirq/TensorFlow Quantum or TensorNetwork/TensorCircuit.Source: §4 · p. 6

Quantum error mitigation

• DQAS inserts single-qubit gates into idle slots to maximize noisy-circuit output fidelity under a specified error model.For QFT-3, six placeholders are searched; the model assumes 2% bit-flip errors between gates and about 20% during long idle periods.Source: §4 · p. 7
• With an expanded gate pool, DQAS places correlated T gates in separated QFT-3 gaps, improving fidelity from 0.33 for the basic circuit to 0.6 for the QEM circuit.The figure reports these fidelities relative to the ideal circuit on noisy hardware.Source: §4 · p. 7, Figure 2 · p. 7
• For QFT-4, DQAS searches 12 gaps and finds noise-mitigation architectures with long-range correlations and multiple inserted gates collectively forming identity operations.The reported example includes three inserted gates on qubit 0 and correlated insertions on qubit 2.Source: §4 · p. 7, Figure 3 · p. 8

QAOA ansatz searching

• On an Erdős–Rényi MAXCUT instance, DQAS reaches the exact cut value 12 with a reduced ansatz and nearly 100% one-shot success probability, versus 47% for P = 2 vanilla QAOA.The reduced ansatz uses about the same quantum resources as P = 1 QAOA while outperforming the vanilla P = 2 circuit.Source: §4 · p. 9

• DQAS can be tailored to specific quantum hardware through customized noise models and native gate sets.Source: §5 · p. 9
• The demonstrated applications include state preparation, unitary decomposition, noiseless and noisy hybrid quantum-classical circuit design, and reduced ansatz design.Source: §5 · p. 9
• DQAS reformulates quantum-circuit and hybrid quantum-classical algorithm design as an automated differentiable architecture-search problem.Source: §5 · p. 9
• A related QAS paper independently used random-search and evolutionary methods, and the two works together support QAS for quantum-circuit optimization.Source: §5 · p. 9

• The DQAS glossary summarizes the framework’s main components for differentiable quantum architecture search.Source: A. Glossary for ingredients of DQAS · p. 13, Table I · p. 13
• DQAS represents the super network with a probabilistic model and uses Monte Carlo sampling with score-function gradients for structural variables.Source: B. Connection to DARTS · p. 13
• Unlike DARTS, DQAS requires probabilistic super-network descriptions because quantum-circuit design makes deterministic evaluation of all paths impractical.Source: B. Connection to DARTS · p. 13
• Sampling batches of simpler circuits avoids the expensive near-term implementation of a complex operation composed from many elementary unitaries.Source: B. Connection to DARTS · p. 13

• The DQAS objective is evaluated through a forward pass over the sampled quantum-circuit architecture.Source: C. Derivation on gradients of… · p. 13
• The derivation uses L(k) as shorthand for L(U(k, θ)) and allows an unnormalized probability distribution with normalization factor Z(α).Source: C. Derivation on gradients of… · p. 13
• DQAS evaluates only one sampled path per quantum-circuit simulation, with path choices determined by the underlying probabilistic model.Source: Figure S1 · p. 14
• The gradient with respect to probabilistic model parameters is obtained by directly applying the score-function approach to the objective.Source: C. Derivation on gradients of… · p. 14
• The resulting gradient formula is the one utilized for model-parameter optimization in DQAS.Source: C. Derivation on gradients of… · p. 14

• The authors leave extensive investigation of adjustable DQAS ingredients and their effects to future work.Source: D. General hyperparameters for DQAS… · p. 14
• Because each DQAS evaluation epoch is expensive, improved learning schedules may be needed to increase training efficiency.Source: D. General hyperparameters for DQAS… · p. 14
• Score-function estimators generally have high variance, and batch sizes of O(100) perform well in circuit-structure searching.Source: D. General hyperparameters for DQAS… · p. 14
• There is no theoretical guarantee that a running-average objective is the best variance-reduction baseline, motivating alternative baselines and variance-control methods.Source: D. General hyperparameters for DQAS… · p. 14
• Different encoding schemes matter in QAS, motivating domain-specific and expressive layer or block encodings beyond simple gate sets.Source: D. General hyperparameters for DQAS… · p. 14
• More sophisticated probabilistic models could represent correlations between circuit layers better than the independent categorical model used here.Source: D. General hyperparameters for DQAS… · p. 15
• Regularization and reward terms could address multiple objectives, including hardware restrictions and quantum-noise reduction.Source: D. General hyperparameters for DQAS… · p. 15
• The listed hyperparameters are not guaranteed to be optimal because the authors did not conduct an extensive hyperparameter search.Source: D. General hyperparameters for DQAS… · p. 15

• Multiple starts train instances with varied data fractions, initialization, or randomization, returning the candidate with the best objective value.Source: E. Training techniques implemented in… · p. 15
• Top-k grid search retains the k most probable operations per layer, producing p^k candidate circuits that can be evaluated to avoid local optima.The usual choice is k = 2.Source: E. Training techniques implemented in… · p. 15
• Additional techniques include parameter noise, regularization, early stopping, score-function baselines, and progressive layer-by-layer training.These techniques address rugged landscapes, overfitting, gradient variance, and difficulty training deep circuits.Source: E. Training techniques implemented in… · pp. 15–16
• Structural regularization penalizes consecutive identical parametrized gates that could be merged, promoting more resource-efficient architectures.For example, rx(θ1)rx(θ2) can be merged into one layer.Source: E. Training techniques implemented in… · p. 16
• Penalties can encourage architectures with fewer two-qubit gates because these gates are identified as major sources of infidelity and computational errors.Source: E. Training techniques implemented in… · p. 16
• DQAS can adapt proxy-task and transfer-learning ideas from DARTS, while broader NAS-inspired techniques are proposed to make searches more robust and versatile.Source: E. Training techniques implemented in… · p. 16

State preparation

• DQAS searches for GHZ-state preparation circuits from |0^n⟩ while progressively reducing depth until the objective can no longer be achieved.The operation pool includes parameterized Ry gates and CNOT gates subject to available connectivity.Source: F. DQAS application in state… · p. 17
• The GHZ objective uses final-state distance, and DQAS tunes Ry(θ) to approximate a Hadamard when Hadamard is absent from the operation pool.Source: F. DQAS application in state… · p. 17

Unitary decomposition

• Unitary decomposition evaluates transformations on all input states rather than only the action on |0^n⟩.Source: F. DQAS application in state… · p. 17

Circuit compiling

• For Bell-state circuit search, DQAS restricts the pool to parameter-free discrete gates and constructs the objective from specified input/output relations.Source: F. DQAS application in state… · p. 17
• The resulting Bell-state circuit has depth p = 5 and is presented as the circuit obtained through DQAS.Source: F. DQAS application in state… · p. 17

State preparation

• The GHZ3 experiment uses Ry and nearest-neighbor CNOT operations, batch size 128, zero circuit-parameter initialization, and Adam with learning rate 0.15 for α.Source: G. Hyperparameter settings and… · p. 18

Bell circuit

• The Bell experiment uses parameter-free X, Y, H, and CNOT operators on each qubit, batch size 128, and Adam with learning rate 0.15 for α.Source: G. Hyperparameter settings and… · p. 18

Quantum error mitigation

• QEM operation pools contain discrete gates including X, Y, Z, S, T, and identity, with the identity representing an idle qubit state.The QFT-3 setup uses batch size 256.Source: G. Hyperparameter settings and… · p. 18
• QEM maximizes fidelity between noisy and ideal outputs, evaluating each circuit on a sampled input because fidelity variation across inputs was numerically small.Source: G. Hyperparameter settings and… · p. 18
• QFT-4 search uses p = 12 slots, but its large search space can trap optimization locally; restricting single-qubit gates can reduce multistart requirements.Source: G. Hyperparameter settings and… · p. 18

QAOA

• QAOA searches use small, large, and extra-large operation pools, including rotation, ZZ, H, XX, YY, and next-nearest-neighbor layers.Source: G. Hyperparameter settings and… · p. 18
• Reduced-graph ZZ layers often use 8–12 subgraphs, but larger graphs or more qubits may require more such layers.Source: G. Hyperparameter settings and… · p. 18
• QAOA experiments favor large batches, Gaussian parameter noise with standard deviation 0.2, and CVaR objectives among tested training ingredients.Source: G. Hyperparameter settings and… · p. 18

Comparison

• The comparison includes bare, naive, and nontrivial DQAS QEM circuits, as well as circuits inspired by theoretical methods.Source: H. Further results on QEM of QFT circuit · p. 19

QFT-3

• QFT-3 fidelity rises from 0.33 for the textbook circuit to 0.55 with naive Pauli-pair insertion and 0.6 with DQAS.Source: H. Further results on QEM of QFT circuit · p. 19

QFT-4

• DQAS discovers multiple QFT-4 QEM configurations with fidelities usually ranging from 0.45 to 0.46, outperforming the human-designed filling policies.Source: H. Further results on QEM of QFT circuit · p. 20
• DQAS also finds distinct QFT-4 circuits with fidelity similar to the optimal circuit reported in the main text.Source: H. Further results on QEM of QFT circuit · p. 20, Figure S5 · p. 20

Hardware-aware search

• DQAS incorporates hardware constraints by translating circuit layers into native gates and adding resource penalties to the search objective.For the stated gate costs, an xx-layer on a degree-3 regular graph has weight 27/2, with weights adjustable as hyperparameters.Source: §I · p. 20

Search-space design

• Block encoding combines primitive operations into multi-layer blocks, helping DQAS discover the standard QAOA layout for P = 3.Examples include zz-rx-block and yy-rz-block constructions while retaining the Hadamard layer in the operation pool.Source: §I · pp. 20–21

Alternative layouts

• For an n = 8 degree-3 regular-graph ensemble, DQAS found an H, yy, rx, zz, rx architecture with MAX CUT expectation around 8.8, similar to same-depth QAOA.The result demonstrates that DQAS can identify competitive layouts beyond the vanilla QAOA structure.Source: §I · p. 21

Ensemble learning

• DQAS rediscovered the QAOA layout for Erdős–Rényi graph ensembles with n = 10, p = 0.3 and n = 8, p = 0.4.The study notes that Erdős–Rényi ensembles usually have greater variance in exact MAX CUT values.Source: §I · p. 21

Instance learning

• For individual MAXCUT instances, DQAS produced a yy, zz, yy circuit with expected energy −8.0 versus −7.75 for same-depth vanilla QAOA.Customized architectures may outperform generic QAOA when circuit depth is restricted, although similar-performing candidates can create local-minimum traps.Source: §I · p. 22