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Differentiable Quantum Architecture Search

Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, Hong Yao

arXiv:2010.08561v2quant-ph

TL;DR

NISQ quantum-circuit design requires adapting architectures to task, noise, and hardware constraints, while existing QAS methods search discrete spaces. DQAS addresses this by jointly optimizing sampled circuit structures and variational parameters in a differentiable probabilistic framework. Across circuit design applications, it discovers task-specific architectures, including noise-resilient circuits and improved QAOA layouts.

  • Problem

    Fixed ansätze and discrete QAS methods limit automated circuit design under NISQ constraints involving noise, depth, connectivity, and problem structure.

  • Method

    DQAS samples circuit structures from a probabilistic model and jointly optimizes its continuous parameters with circuit variational parameters through gradient-based differentiation.

  • Results

    DQAS designs circuits across decomposition, error mitigation, and optimization tasks, including a MAX CUT circuit achieving 12 versus 10.39 for P = 1 vanilla QAOA and 11.18 for P = 2 vanilla QAOA.

  • Takeaways & Limitations

    DQAS provides a flexible framework for customized quantum-circuit design incorporating native gates, hardware connectivity, and error models.

  • Takeaways & Limitations

    The presented probabilistic model is independent across categories and does not explicitly model correlations between circuit layers; richer models remain future work.

Abstract

from arXiv · show

Quantum architecture search (QAS) is the process of automating architecture engineering of quantum circuits. It has been desired to construct a powerful and general QAS platform which can significantly accelerate current efforts to identify quantum advantages of error-prone and depth-limited quantum circuits in the NISQ era. Hereby, we propose a general framework of differentiable quantum architecture search (DQAS), which enables automated designs of quantum circuits in an end-to-end differentiable fashion. We present several examples of circuit design problems to demonstrate the power of DQAS. For instance, unitary operations are decomposed into quantum gates, noisy circuits are re-designed to improve accuracy, and circuit layouts for quantum approximation optimization algorithm are automatically discovered and upgraded for combinatorial optimization problems. These results not only manifest the vast potential of DQAS being an essential tool for the NISQ application developments, but also present an interesting research topic from the theoretical perspective as it draws inspirations from the newly emerging interdisciplinary paradigms of differentiable programming, probabilistic programming, and quantum programming.

1. INTRODUCTION

Quantum architecture search automates quantum-circuit design, addressing the limitations of fixed variational ansätze and NISQ hardware. DQAS introduces a general, end-to-end differentiable framework inspired by differentiable neural architecture search.

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

2. BACKGROUND AND RELATED WORK

Prior neural and quantum architecture-search methods largely operate over discrete spaces, motivating differentiable search. The QAOA background supplies a structured target for discovering generalized Hamiltonian layouts beyond the vanilla ansatz.

  • Differentiable Neural Architecture Search: NAS seeks task-specific high-performing neural architectures using reinforcement learning, evolutionary algorithms, or differentiable relaxation.
  • Differentiable Neural Architecture Search: DARTS relaxes discrete architecture components into continuous variables optimized by differentiation and gradient descent, reducing training time and hardware requirements.
  • Related works on QAS: Earlier QAS studies mainly used evolutionary, greedy, or reinforcement-learning methods for QAOA and VQE-related circuit-structure tasks.
  • Related works on QAS: These QAS methods search discrete quantum architectures, motivating DQAS as an end-to-end differentiable alternative.
  • Brief review on QAOA: QAOA alternates phase or cost evolution with mixer evolution, using variational parameters to address combinatorial optimization problems.
  • Brief review on QAOA: Generalized QAOA separates preparation, mixer, phase/problem, and cost Hamiltonians, creating layouts that DQAS can automatically search beyond vanilla QAOA.
  • Brief review on QAOA: QAOA uses relatively few variational parameters, potentially easing training compared with same-depth hardware-efficient VQE or quantum neural-network ansätze.

3. METHODS

DQAS jointly searches circuit structure and variational parameters by sampling discrete architectures from a differentiable probabilistic model. It supports task-specific objectives, broad circuit-design settings, and scalable parameter sharing.

  • Overview: DQAS treats circuit design as bi-optimization: it selects unitaries for a circuit layout while identifying suitable variational parameters.
  • 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.
  • Circuit encoding and operation pool: Each placeholder can reuse an operation-pool unitary, allowing DQAS to assemble circuits with replacement.
  • Objectives: Objectives cover hybrid quantum-classical tasks, supervised learning, state preparation, and circuit compilation, including expectation-based losses and reference-circuit fidelity.
  • 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.
  • 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.
  • Probabilistic models: The default independent-category probabilistic model omits explicit inter-layer correlations; richer energy-based or autoregressive models are left for future work.

4. APPLICATIONS

DQAS is applied to several NISQ circuit-design tasks, including noise-resilient QFT circuits and QAOA-like MAXCUT ansatz search. It discovers correlated gate insertions, circuit layouts, and reduced ansatz structures automatically.

  • 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.
  • 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.
  • Quantum error mitigation: 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.
  • Quantum error mitigation: 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.
  • 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.

5. DISCUSSIONS

DQAS is presented as a versatile framework for automated, hardware-aware quantum circuit design across several NISQ applications. The work highlights reduced ansatz search, including shallower QAOA circuits that outperform conventional layouts, while noting related QAS efforts.

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

A. Glossary for ingredients of DQAS

DQAS uses a probabilistic super-network representation and Monte Carlo score-function gradients to search quantum circuits with implementable sampled unitary circuits. This probabilistic representation is more than an optional memory-saving device because direct super-network implementation is costly on NISQ hardware.

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

C. Derivation on gradients of probabilistic model parameters

The gradient derivation defines the objective’s forward-pass evaluation and applies a score-function approach to obtain gradients for probabilistic model parameters. The derivation concludes with the gradient formula used in DQAS.

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

D. General hyperparameters for DQAS training.

DQAS training depends on choices for optimization, batching, encoding, probabilistic modeling, regularization, and parameter reuse. The authors identify these ingredients as adjustable and leave extensive investigation of their effects to future work, without guaranteeing optimal hyperparameters.

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

E. Training techniques implemented in DQAS

DQAS training can incorporate techniques adapted from differentiable neural architecture search to improve robustness, reduce redundancy, control resources, and explore complex landscapes.

  • Multiple starts train instances with varied data fractions, initialization, or randomization, returning the candidate with the best objective value.
  • 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.
  • 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.
  • 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.
  • Penalties can encourage architectures with fewer two-qubit gates because these gates are identified as major sources of infidelity and computational errors.
  • 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.

F. DQAS application in state preparation and circuit compiling

DQAS designs circuits for state preparation and unitary compilation by optimizing task-specific objectives over candidate quantum structures. Examples include GHZ-state preparation and a depth-5 Bell-state circuit.

  • 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.
  • State preparation: The GHZ objective uses final-state distance, and DQAS tunes Ry(θ) to approximate a Hadamard when Hadamard is absent from the operation pool.
  • Unitary decomposition: Unitary decomposition evaluates transformations on all input states rather than only the action on |0^n⟩.
  • 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.
  • Circuit compiling: The resulting Bell-state circuit has depth p = 5 and is presented as the circuit obtained through DQAS.

G. Hyperparameter settings and training ingredients in experiments

The experiments vary operation pools, objectives, noise, batch sizes, and search strategies across state preparation, Bell circuits, QEM, and QAOA applications.

  • 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.
  • 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.
  • 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.
  • Quantum error mitigation: QEM maximizes fidelity between noisy and ideal outputs, evaluating each circuit on a sampled input because fidelity variation across inputs was numerically small.
  • Quantum error mitigation: QFT-4 search uses p = 12 slots, but its large search space can trap optimization locally; restricting single-qubit gates can reduce multistart requirements.
  • QAOA: QAOA searches use small, large, and extra-large operation pools, including rotation, ZZ, H, XX, YY, and next-nearest-neighbor layers.
  • QAOA: Reduced-graph ZZ layers often use 8–12 subgraphs, but larger graphs or more qubits may require more such layers.
  • QAOA: QAOA experiments favor large batches, Gaussian parameter noise with standard deviation 0.2, and CVaR objectives among tested training ingredients.

H. Further results on QEM of QFT circuit

DQAS-designed QEM circuits improve fidelity over bare and human-designed QEM circuits for QFT-3 and QFT-4 under the described bit-flip-noise setting.

  • Comparison: The comparison includes bare, naive, and nontrivial DQAS QEM circuits, as well as circuits inspired by theoretical methods.
  • 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.
  • 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.
  • QFT-4: DQAS also finds distinct QFT-4 circuits with fidelity similar to the optimal circuit reported in the main text.

I. Further results on QAOA ansatz searching

Further results show DQAS can recover standard QAOA layouts, discover competitive alternatives, and customize circuits for individual MAXCUT instances. The study also examines hardware-aware objectives, search strategies, transfer to larger systems, and limitations from rugged landscapes and objective choice.

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
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