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arXiv:2610.02029

(A Variant of) Clifford Circuit Synthesis is NP-Complete

Luna Lima Keller, Richard Kueng

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

Optimal circuit synthesis is the problem of finding the shortest-depth circuit representation of a given functionality with respect to a pre-specified elementary gate set. This is a central problem in both quantum and classical hardware design. While the classical version is very well understood -- both in terms of heuristics and rigorous hardness assertions -- much less is known about optimal quantum circuit synthesis. We focus on optimal Clifford circuit synthesis, for which various heuristics are known, e.g. via reduction to 3-SAT. Our main result supplies a matching hardness result: a variant of optimal Clifford synthesis is NP-hard. The proof proceeds in two parts: (i) reduce 3-edge colorability on 3-regular graphs to a circuit synthesis problem that only involves CZ gates, (ii) prove that the availability of additional elementary Clifford gates -- most notably: Hadamard, phase and CNOT -- cannot lead to further improvements of the optimal circuit depth. Our work sharpens the complexity-theoretic understanding of Clifford circuits: simulation and equivalence checking are in P, whereas deciding whether a Clifford unitary admits an implementation within a prescribed depth is NP-complete.

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Optimal circuit synthesis is the problem of finding the shortest-depth circuit representation of a given functionality with respect to a pre-specified elementary gate set. This is a central problem in both quantum and classical hardware design. While the classical version is very well understood -- both in terms of heuristics and rigorous hardness assertions -- much less is known about optimal quantum circuit synthesis.
We focus on optimal Clifford circuit synthesis, for which various heuristics are known, e.g. via reduction to 3-SAT. Our main result supplies a matching hardness result: a variant of optimal Clifford synthesis is NP-hard. The proof proceeds in two parts: (i) reduce 3-edge colorability on 3-regular graphs to a circuit synthesis problem that only involves CZ gates, (ii) prove that the availability of additional elementary Clifford gates -- most notably: Hadamard, phase and CNOT -- cannot lead to further improvements of the optimal circuit depth. Our work sharpens the complexity-theoretic understanding of Clifford circuits: simulation and equivalence checking are in P, whereas deciding whether a Clifford unitary admits an implementation within a prescribed depth is NP-complete.
We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.
Minimizing the number of CNOT gates required to synthesize a Clifford operator is a central problem in quantum circuit optimization. We extend the link--middle--cut (LMC) framework for linear reversible circuits to Clifford operators represented by binary symplectic tableaux. By introducing block-support connectivity graphs and determinantal invariants, we obtain efficiently computable lower bounds on ancilla-free CNOT complexity. We prove that these bounds are tight for qubit permutations: a permutation of nnn qubits with kkk cycles requires exactly 3(n−k)3(n-k)3(n−k) CNOT gates, even when arbitrary one-qubit Clifford gates are available. We also derive efficiently computable bounds for synthesis up to a permutation of the output qubits, corresponding to free qubit relabeling. Finally, we further demonstrate the utility of the bound by using it as an admissible heuristic for A∗A^*A∗-based Clifford synthesis and show that, on benchmark instances, the resulting search can prove optimality beyond what was possible with earlier SAT-based methods, and that it may be used in combination with earlier A∗A^*A∗ heuristics to improve the CNOT counts obtained through heuristic search.
Efficiently implementing Clifford circuits is crucial for quantum error correction and quantum algorithms. Linear reversible circuits, equivalent to circuits composed of CNOT\xspace gates, have important applications in classical computing. In this work we present methods for CNOT\xspace and general Clifford circuit synthesis which can be used to minimise either the entangling two-qubit gate count or the circuit depth. We present three families of algorithms - optimal synthesis which works on small circuits, A* synthesis for intermediate-size circuits and greedy synthesis for large circuits. We benchmark against existing methods, including rustiq, tket and qiskit and show that our approach results in circuits with lower two-qubit gate count. For encoding circuits, our methods outperform previous reinforcement learning results and find a lower two-qubit gate count circuit for the Golay code than previously known. The algorithms have been implemented in a GitHub repository for use by the classical and quantum computing community.
Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of 2n2n2n Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an nnn-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in O(n2)\mathcal{O}\left(n^2\right)O(n2) time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to n=80n=80n=80 qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements ×2.57\times2.57×2.57 and ×2.28\times2.28×2.28 in the total number of gates, respectively.
In fault-tolerant quantum circuit synthesis, T gates supplied via magic states dominate space-time cost, while Clifford gates incur negligible overhead. Conventional flows minimize AND count in an {XOR, AND, NOT} basis as a proxy for T, which neglects phase cancellation and can be far from T-optimal. We instead formulate an exact T synthesis problem and canonicalize Boolean functions under Clifford equivalence. By precomputing T-optimal implementations up to seven variables and developing a customized mapper, we reduce the T count by up to 14.3% on EPFL benchmarks and improve the T counts of several cryptographic modules by up to 40%.
Path sums are a convenient symbolic formalism for quantum operations with applications to the simulation, optimization, and verification of quantum protocols.Unlike quantum circuits, path sums are not limited to unitary operations, but can express arbitrary linear ones.Two problems, therefore, naturally arise in the study of path sums: the unitarity problem and the extraction problem.The former is the problem of deciding whether a given path sum represents a unitary operator.The latter is the problem of constructing a quantum circuit, given a path sum promised to represent a unitary operator.In this paper, we show that the unitarity problem is co-NP-hard in general, but that it is in P when restricted to Clifford path sums.We then provide an algorithm to synthesize a Clifford circuit from a unitary Clifford path sum.The circuits produced by our extraction algorithm are of the form C 1 HC 2 , where C 1 and C 2 are Hadamard-free circuits and H is a layer of Hadamard gates.We also provide a heuristic generalization of our extraction algorithm to arbitrary path sums.While this algorithm is not guaranteed to succeed, it often succeeds and typically produces natural looking circuits.Alongside applications to the optimization and decompilation of quantum circuits, we demonstrate the capability of our algorithm by synthesizing the standard quantum Fourier transform directly from a path sum.