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AlphaClifford: Efficient Clifford Synthesis and Transpilation with Model-based RL

Daniele Lizzio Bosco, Jacopo Cossio, Carla Piazza, Giuseppe Serra

arXiv:2608.18946v1quant-phcs.AI

TL;DR

Clifford circuit synthesis and transpilation require better optimization than standard heuristics provide. AlphaClifford uses model-based reinforcement learning with Monte Carlo Tree Search over symplectic-matrix states, achieving up to 56% total-gate reduction versus the Aaronson-Gottesman algorithm and strong performance across additional compilation tasks.

  • Problem

    Standard Clifford compilation methods can produce sub-optimal circuits, motivating improved synthesis and transpilation approaches for this central gate set.

  • Method

    AlphaClifford applies model-based reinforcement learning and Monte Carlo Tree Search over the algebraic space of symplectic matrices for Clifford synthesis and transpilation.

  • Results

    Up to 56% total gate reduction versus the Aaronson-Gottesman algorithm was achieved, with comparable or superior performance across unconstrained optimization, hardware-constrained transpilation, and logical-synthesis post-optimization.

  • Takeaways & Limitations

    AlphaClifford provides a broadly applicable model-based approach for optimizing Clifford compilation across unconstrained, hardware-constrained, and logical-synthesis settings.

Abstract

from arXiv · show

Clifford circuits play a foundational role in quantum computing, particularly due to their importance in quantum error correction and fault-tolerant logical synthesis. While these circuits can be efficiently simulated and represented as symplectic matrices, standard synthesis methods-such as the Aaronson-Gottesman algorithm-often yield sub-optimal circuits with excessively high gate counts. In this work, we introduce AlphaClifford, a model-based Reinforcement Learning framework powered by Monte Carlo Tree Search, designed to efficiently synthesize Clifford circuits from the fundamental gate set composed of H, S, and CNOT. By modeling the state space through the algebraic properties of the symplectic group, AlphaClifford effectively explores this combinatorial space to minimize overall circuit cost. For unconstrained Clifford optimization, our approach achieves a consistent reduction in both total and two-qubit (CNOT) gate counts compared to state-of-the-art synthesis heuristics, despite operating with a strictly less expressive gate set. Furthermore, we demonstrate the broad applicability of our framework on two additional tasks: hardware-constrained Clifford transpilation, where we outperform existing RL-based compilers, and as a post-synthesis optimization component within a full Clifford+T logical synthesis pipeline. Our results underscore that model-based RL is highly effective at addressing the combinatorial complexities of quantum compilation, offering a scalable pathway to mitigate hardware constraints in both near-term and future fault-tolerant quantum devices.

I. INTRODUCTION

AlphaClifford applies reinforcement learning to Clifford-circuit synthesis and transpilation using symplectic-matrix representations and the H, S, CNOT gate set. Across three tasks, it reports substantial reductions relative to established synthesis and transpilation methods.

  • Motivation: Quantum algorithm execution is constrained by circuit depth and total physical-gate count, motivating quantum circuit optimization.Equivalent unitary transformations can be implemented by different gate sequences, making synthesis a search for an efficient sequence.
  • Scope: Clifford circuits generated by H, S, and CNOT are central to logical synthesis, particularly within Clifford+T decomposition.The work addresses both synthesis and transpilation for this gate set.
  • Method: AlphaClifford uses a compact symplectic-matrix representation and tree-based Monte Carlo search to learn efficient Clifford syntheses and transpilation strategies.The search seeks a shortest path from an initial matrix to the identity, reconstructing circuits from H, S, and CNOT.
  • Evaluation: The method is evaluated on unconstrained Clifford optimization, hardware-connectivity-constrained transpilation, and refinement within a full logical synthesis pipeline.These tasks are presented as demonstrations of the framework’s broad applicability.
  • Results: Up to 56% total gate reduction is achieved versus Aaronson-Gottesman, while up to 20% improvement is reported over an RL-based Clifford transpiler for transpilation.The introduction also characterizes performance as comparable or higher than a variety of techniques.

II. SYMPLECTIC REPRESENTATION OF CLIFFORD CIRCUITS · A. Stabilizer Formalism and Tableau Representation

Clifford circuits normalize the Pauli group and are generated by H, S, and CNOT, while their evolution can be efficiently simulated using stabilizer tableaus. The tableau’s binary X and Z components encode Pauli action, and its phase vector records accumulated generator signs.

  • II. SYMPLECTIC REPRESENTATION OF CLIFFORD CIRCUITS: Clifford operators conjugate every Pauli operator to another Pauli operator and are finitely generated by H, S, and CNOT.This normalizer structure defines the Clifford group within U(2^n).
  • A. Stabilizer Formalism and Tableau Representation: Clifford circuits are efficiently classically simulable, with Aaronson–Gottesman tableaus tracking n destabilizer and n stabilizer generators.The tableau is a 2n×(2n+1) binary matrix over F2.
  • A. Stabilizer Formalism and Tableau Representation: The tableau’s 2n×n Boolean X and Z matrices record the presence of X and Z Pauli operators across the generators.Together, these matrices describe the circuit’s Pauli structure.
  • A. Stabilizer Formalism and Tableau Representation: For one qubit, the identity tableau contains destabilizer X in its first row and stabilizer Z in its second row.This provides the basic tableau initialization used to illustrate Clifford transformations.
  • A. Stabilizer Formalism and Tableau Representation: Applying H exchanges X and Z, demonstrating that Clifford operations act linearly on tableau Pauli components.Applying X afterward leaves the Pauli structure unchanged but flips the Z generator’s phase.
  • A. Stabilizer Formalism and Tableau Representation: The X and Z matrices encode Pauli action, whereas r captures the circuit’s accumulated phase information.This separates structural Pauli evolution from generator-sign tracking.

B. Symplectic Matrix Representation

Clifford circuits’ entangling structure and basis transformations are represented by a 2n × 2n binary symplectic matrix M, while Pauli phases require a separate phase vector. This representation places M in Sp(2n, F2) and arises from a surjective map from Clifford unitaries whose kernel is the Pauli group.

  • Matrix representation: The 2n × 2n binary matrix M = [X Z] captures Clifford circuits’ entangling structure and basis transformations, whereas the phase vector r tracks exact Pauli phases.The matrix representation does not encode the full quantum state’s Pauli phases.
  • Symplectic structure: M must satisfy the symplectic condition over F2 to preserve Pauli-group commutation relations, making it an element of Sp(2n, F2).The symplectic form uses a 2n × 2n block matrix built from the n × n identity matrix I_n.
  • Group homomorphism: A surjective group homomorphism ϕ : C_n → Sp(2n, F2) maps each Clifford unitary to its symplectic matrix, with kernel P_n when global phases are ignored.The kernel is precisely the Pauli group under this phase convention.

C. Implications for Circuit Synthesis · III. RELATED WORK

The paper frames Clifford synthesis through a compact symplectic representation that preserves circuit equivalence up to a Pauli layer, while positioning AlphaClifford among heuristic, exact, model-free, and model-based approaches. AlphaClifford extends search-informed reinforcement learning to general Clifford-circuit synthesis by exploiting Clifford algebraic structure.

  • C. Implications for Circuit Synthesis: Matching a target symplectic matrix with {H, S, CNOT} guarantees the synthesized and target circuits lie in the same Pauli-group coset.The equivalence is up to a Pauli layer.
  • C. Implications for Circuit Synthesis: The residual Pauli correction P can be computed in O(n2) time by tracking the phase vector r during synthesis.This enables efficient correction after evolving the symplectic representation.
  • C. Implications for Circuit Synthesis: Mapping the RL state space to Sp(2n, F2) instead of the full Clifford group Cn reduces the search space by a factor of 22n.The reduction simplifies the learning process.
  • III. RELATED WORK: The Aaronson-Gottesman algorithm is the historical baseline, efficiently simulating and synthesizing Clifford circuits in polynomial time using {H, S, CNOT}.Its standard approaches nevertheless tend to produce sub-optimal gate counts.
  • III. RELATED WORK: Heuristic alternatives use richer generating sets, while Clifford-specific template matching and symbolic peephole optimization reduce circuit gate counts, particularly CNOT counts.Examples include A*, greedy algorithms, Volanto, and the approach described in.
  • III. RELATED WORK: Exact minimum-gate Clifford synthesis generally requires exponential time, limiting scalability to very small numbers of qubits and practical applicability.The literature also includes optimal and exact synthesis methods.
  • III. RELATED WORK: Model-free RL methods have addressed Clifford routing, synthesis, state preparation, architecture search, and logical-synthesis pre-compilation, while model-based MCTS methods better explore structured combinatorial spaces.Prior examples include PPO, AlphaTensor-Quantum, AlphaCNOT, and QuSynth.
  • III. RELATED WORK: AlphaClifford extends search-informed RL to general Clifford-circuit synthesis by adapting AlphaTensor and AlphaCNOT ideas to Clifford operations’ algebraic structure.The work builds on prior developments to advance Clifford-circuit synthesis.

IV. ALPHACLIFFORD

AlphaClifford is a model-based RL framework that explores the combinatorial space of symplectic matrices associated with Clifford circuits. It models this space as a search tree, trains a model using a Monte Carlo Tree Search pattern, and synthesizes equivalent circuits by generating gate sequences.

  • Motivation: Model-based RL represents the environment to simulate future outcomes, enabling structured exploration of complex combinatorial spaces such as symplectic matrices.This contrasts with model-free approaches such as PPO, which learn directly from state–action–reward interactions without an explicit dynamics model.
  • Approach: AlphaClifford applies this model-based principle to explore the combinatorial space of symplectic matrices associated with Clifford circuits.The approach is inspired by recent applications including AlphaTensor and AlphaCNOT.
  • Search-space modeling: The method models symplectic matrices as search-tree nodes, with directed edges representing Clifford-gate applications between reachable matrices.Each node corresponds to a specific symplectic matrix, and connected nodes differ by a Clifford gate operation.
  • Training and synthesis: A trained model predicts gate sequences transforming an initial symplectic matrix into a target one, following a Monte Carlo Tree Search training pattern.After training, it generates Clifford-gate sequences whose symplectic matrix matches that of the target circuit.

A. Modeling the problem of symplectic matrices · B. Inverting the representation: from the target to the identity

AlphaClifford models Clifford synthesis as tree search over symplectic matrices, with admissible H, S, and CNOT operations defining transitions toward the target. An inverted representation embeds the target in the initial state and converts synthesis into reaching a fixed identity state with reversed dynamics.

  • A. Modeling the problem of symplectic matrices: Clifford synthesis searches for a gate sequence equivalent to target matrix M_C ∈ Sp(2n, F_2), optionally respecting a connectivity map M.The problem dimension n corresponds to the number of qubits.
  • A. Modeling the problem of symplectic matrices: The search tree roots at identity I_2n, representing the empty circuit, and expands through admissible H_i, S_i, and CNOT_ij operations.Each child applies the corresponding gate matrix to its parent’s symplectic matrix.
  • A. Modeling the problem of symplectic matrices: Terminal nodes represent target matrix M_C; unconstrained nodes have Θ(n^2) children, while connectivity constraints restrict the allowed two-qubit operations.The branching factor reflects possible CNOT operations between qubit pairs together with single-qubit gates.
  • B. Inverting the representation: from the target to the identity: The forward representation would provide both current matrix M_k and fixed target M_C at every step, increasing input dimensionality while adding little episode-specific information.The target remains constant throughout an episode.
  • B. Inverting the representation: from the target to the identity: The inverted formulation embeds the target directly into the observation, representing the inverse of the current symplectic matrix and corresponding inverse circuit.This reformulates synthesis through the reversed circuit representation.
  • B. Inverting the representation: from the target to the identity: Each episode starts at M_C and seeks the identity matrix, turning synthesis into goal-reaching with a fixed terminal state and one observation matrix.The model receives a single matrix instead of the pair (M_k, M_C).
  • B. Inverting the representation: from the target to the identity: Although the agent selects forward Clifford gates, inverted-state evolution uses the corresponding adjoint action and right-multiplies by their symplectic matrices.This update consistently represents the reversed dynamics of circuit construction.

C. Monte Carlo Tree Search Framework · D. Training Details

AlphaClifford uses MCTS over symplectic-matrix states to synthesize Clifford circuits with UCT-guided exploration, expansion, and topology-aware action constraints. Training samples random targets, rewards shorter identity-reaching decompositions, and progressively increases instance difficulty through a 95%-success curriculum.

  • C. Monte Carlo Tree Search Framework: MCTS explores the symplectic-matrix space associated with Clifford circuits through four structured phases.The framework starts from the target symplectic matrix and searches for a sequence of admissible Clifford gates.
  • C. Monte Carlo Tree Search Framework: Selection traverses the tree from the target matrix using UCT to balance exploration of less-visited actions with exploitation of high-value actions.Actions are Clifford gates from {H, S, CNOT}, and UCT uses estimated action values and visit counts.
  • C. Monte Carlo Tree Search Framework: Expansion applies unexplored admissible Clifford operations to non-terminal nodes, while termination occurs at the identity matrix or when stopping criteria such as maximum depth are met.Under connectivity constraints, only allowed CNOT operations are considered.
  • D. Training Details: Each training episode generates a different random target and asks the model to reach identity from its inverse using the fewest feasible gate moves.Feasible moves are H, S, and CNOT gates, potentially restricted by topology connectivity.
  • D. Training Details: Episodes end upon reaching identity or after nmax moves without reaching it.The move limit provides a termination condition for unsuccessful episodes.
  • D. Training Details: The reward applies −1/√n per step and +n upon reaching identity, implicitly minimizing total applied gates without problem-specific heuristics.The paper describes this non-informed reward as allowing the model to learn shorter decompositions naturally.
  • D. Training Details: Training uses a linear curriculum in which target matrices are generated from random k-gate Clifford sequences, with k increased after a solving rate of at least 95%.Training starts at k = 1 and stops when the maximum difficulty kmax = 4n2 is reached.

E. Inference … 2) Transpilation:

AlphaClifford infers equivalent Clifford circuits by synthesizing gate sequences from target symplectic matrices and correcting phase discrepancies. The framework supports optimization, hardware-constrained transpilation, and integration into logical synthesis pipelines.

  • E. Inference: Inference synthesizes a gate sequence Csyn whose symplectic matrix matches the target, Mtar = Msyn, making the circuits equivalent up to a Pauli operation.The model can optionally synthesize under hardware connectivity constraints.
  • E. Inference: The complete inference procedure computes Mtar, queries the trained model, evaluates stabilizer and destabilizer phase discrepancies, and applies X and Z corrections.These corrections yield a fully equivalent circuit.
  • F. Applications: The framework targets three applications: Clifford optimization, Clifford transpilation, and use as a subroutine in a standard logical synthesis pipeline.These applications correspond to the experimental sections V-A, V-B, and V-C.
  • 1) Optimization:: Clifford optimization seeks an equivalent representation minimizing the total number of two-qubit gates.The model can also be trained to optimize alternative metrics such as overall circuit depth.
  • 1) Optimization:: AlphaClifford can act as a peephole optimizer by extracting contiguous subcircuits with limited qubit support from larger synthesized Clifford circuits.The passage gives Aaronson–Gottesman synthesis as an example starting point.
  • 2) Transpilation:: Hardware-constrained transpilation transforms a logical circuit into an equivalent physical circuit that obeys a connectivity graph’s allowable direct interactions.This requires mapping logical qubits to physical qubits and inserting additional gates.
  • 2) Transpilation:: Transpilation minimizes a chosen cost function, typically two-qubit gate count or overall circuit depth, while optionally penalizing high-error qubits or couplers.The framework enforces hardware constraints by restricting the model’s action space during training.

3) Optimizing Logical Synthesis Pipeline: · V. EXPERIMENTAL RESULTS · A. Clifford Optimization

AlphaClifford is evaluated as both a post-synthesis optimizer for Clifford blocks in Clifford+T pipelines and a full Clifford synthesizer across varying qubit sizes. In Clifford optimization, it generally achieves the lowest total and two-qubit gate counts despite using the less expressive {H, S, CNOT} gate set.

  • 3) Optimizing Logical Synthesis Pipeline:: Logical synthesis approximates target unitaries over universal discrete sets such as Clifford+T, but exact optimal synthesis scales exponentially with qubit count.Compilers therefore commonly use peephole synthesis strategies for tractability.
  • 3) Optimizing Logical Synthesis Pipeline:: The standard pipeline preprocesses logical circuits, locally compiles sub-blocks into Clifford+T, and can use AlphaClifford afterward to optimize extracted Clifford blocks.AlphaClifford compresses these blocks while preserving the expensive T-gate layout.
  • 3) Optimizing Logical Synthesis Pipeline:: Although Section V-C evaluates relatively small peephole-generated Clifford blocks, the approach also applies to compilation tasks containing large consecutive Clifford blocks.The same model can optimize larger Clifford regions when they occur in other compilation settings.
  • V. EXPERIMENTAL RESULTS: The experimental study covers Clifford optimization, Clifford transpilation, and optimization within the logical synthesis pipeline.These evaluations are organized as Sections V-A, V-B, and V-C.
  • A. Clifford Optimization: AlphaClifford is compared using average total-gate and two-qubit-gate counts, both one-shot and after 10 samples through the AlphaClifford10 variant.The comparison includes Aaronson-Gottesman and the A∗, greedy, and Volanto heuristics.
  • A. Clifford Optimization: AlphaClifford achieves the lowest total and two-qubit gate counts in most cases, while all methods solve every instance except Greedy on some larger dimensions.Its advantage over the methods in is obtained despite the less expressive {H, S, CNOT} gate set.

B. Transpilation

AlphaClifford transpiles unconstrained random Clifford circuits onto hardware connectivity maps using H, S, and connectivity-limited CNOT gates. Across six maps from 3 to 6 qubits, it is evaluated against Qiskit’s RL-based Clifford transpiler using total and CNOT gate counts under stochastic sampling.

  • Transpilation: The model converts random unconstrained Clifford circuits into equivalent circuits using H, S, and CNOT gates restricted by the target connectivity map.A CNOT may be selected only between qubit pairs allowed by the hardware map.
  • Transpilation: The benchmark uses 6 connectivity maps spanning 3 to 6 qubits, including line and T-shape graphs.Vertices denote physical qubits, while edges denote available coupling-map connections.
  • Transpilation: The evaluation reports total gate count and CNOT count against Qiskit’s RL-based Clifford transpiler.Both methods are sampled once and 10 times to account for the stochasticity of RL-based approaches.
  • Transpilation: Our model generally provides consistently lower total gate counts than the Qiskit RL-based Clifford transpiler.Results are reported in Table II under both one-sample and ten-sample evaluations.

C. Full Logical synthesis pipeline · VI. CONCLUSION

The paper evaluates AlphaClifford within a full logical synthesis pipeline and concludes that model-based reinforcement learning can optimize Clifford synthesis and transpilation across unconstrained, hardware-constrained, and post-synthesis settings. It also identifies expanded actions, hardware-aware rewards, and systematic block-selection strategies as future directions.

  • C. Full Logical synthesis pipeline: The pipeline evaluates 10 random circuits for each qubit count n ∈ {5, 10, 15, 20, 25}.The evaluation follows the approach in.
  • C. Full Logical synthesis pipeline: Two-qubit gates are decomposed into CNOT and single-qubit gates using KAK decomposition, followed by single-qubit decomposition into up to three Rz(θ) rotations.GRIDSYNTH decomposes each rotation into {H, S, T} with tolerance ε = 0.01.
  • C. Full Logical synthesis pipeline: AlphaClifford synthesizes adjacent Clifford blocks of at least 2 qubits, while a parallel pipeline uses Q-PRESYN with greedy two-qubit merges to reduce resulting T gates.The pipeline searches for blocks of adjacent Clifford circuits before applying the respective optimization method.
  • C. Full Logical synthesis pipeline: Table III compares Clifford gate reductions against the default qiskit transpiler, noting that the evaluated random circuits mostly produce 2- and 3-qubit Clifford blocks.Because these Clifford circuits are already optimal in CNOT count, Aaronson-Gottesman and the methods from achieved no total-gate reduction in this setting.
  • VI. CONCLUSION: AlphaClifford formulates compilation as Monte Carlo Tree Search over the algebraic space of symplectic matrices to discover optimized gate sequences.The framework is introduced as model-based reinforcement learning for Clifford circuit synthesis and transpilation.
  • VI. CONCLUSION: AlphaClifford achieves superior or comparable results across unconstrained optimization, hardware-constrained transpilation, and post-synthesis logical optimization.The conclusion specifically reports minimizing total and two-qubit gate counts against standard heuristics despite using a less expressive gate set.
  • VI. CONCLUSION: Model-based reinforcement learning is presented as a way to overcome traditional heuristic-driven compiler limitations as quantum hardware approaches near-term and future physical constraints.The conclusion frames AlphaClifford alongside AlphaTensor and AlphaCNOT and identifies systematic block-selection and iterative peephole strategies as further research directions.
  • VI. CONCLUSION: The authors propose expanding the action space with operations such as transvections to further improve compilation efficiency.They also suggest tailored rewards that penalize two-qubit gates or incorporate error-aware routing for hardware-specific solutions.
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