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arXiv:2610.40886

Clifford synthesis is NP-complete

Søren Fuglede Jørgensen

quant-phcs.CCSubmitted Oct 29, 2026• Announced Nov 2, 2026
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Abstract

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.

AI Summary

The paper asks whether a specified Clifford operator admits an exact implementation within a gate-count budget when qubits have unrestricted connectivity and no additional ancillas are allowed. It extends recent NP-completeness results for CNOT-only synthesis to circuits that may use arbitrary single-qubit Clifford gates. Deciding whether a target can be implemented with at most $k$ CNOTs is NP-complete even when single-qubit Cliffords are free. Hardness already holds for linear reversible targets $\rho(A)=\operatorname{diag}(A,(A^{-1})^\top)$ with $A$ lower unitriangular and $(A-I)^3=0$. The result also holds when arbitrary two-qubit Clifford gates each count as one resource, when SWAPs are free, and when single- and two-qubit gates carry fixed nonnegative and positive rational costs, respectively, provided the two-qubit gate set contains CNOT. The reduction starts from Hamiltonian path in grid graphs and extends a recorder-qubit construction for CNOT synthesis. For a graph with $m$ vertices, it uses $r=6(m-1)+1$ recorders per vertex, $R=mr$ recorders total, $O(m^2)$ qubits, and threshold $k=R+m-1$. Tracking intersections of qubit symplectic planes with a Lagrangian subspace yields appearance-count bounds that prevent single-qubit Clifford shortcuts: any circuit meeting the threshold induces a CNOT-only implementation encoding a Hamiltonian path. Further counting handles output permutations and hence free SWAPs and arbitrary two-qubit Cliffords. Polynomial-size certificates establish membership in NP. The results concern exact gate-count optimization without extra ancillas, not depth optimization. They establish worst-case decision complexity, rather than approximation hardness or typical-instance difficulty.

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Large language models were used during the preparation of the paper, in particular to prepare drafts of some of the proofs.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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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.
The paper asks whether a specified Clifford operator admits an exact implementation within a gate-count budget when qubits have unrestricted connectivity and no additional ancillas are allowed. It extends recent NP-completeness results for CNOT-only synthesis to circuits that may use arbitrary single-qubit Clifford gates.
Deciding whether a target can be implemented with at most kkk CNOTs is NP-complete even when single-qubit Cliffords are free. Hardness already holds for linear reversible targets ρ(A)=diag⁡(A,(A−1)⊤)\rho(A)=\operatorname{diag}(A,(A^{-1})^\top)ρ(A)=diag(A,(A−1)⊤) with AAA lower unitriangular and (A−I)3=0(A-I)^3=0(A−I)3=0. The result also holds when arbitrary two-qubit Clifford gates each count as one resource, when SWAPs are free, and when single- and two-qubit gates carry fixed nonnegative and positive rational costs, respectively, provided the two-qubit gate set contains CNOT.
The reduction starts from Hamiltonian path in grid graphs and extends a recorder-qubit construction for CNOT synthesis. For a graph with mmm vertices, it uses r=6(m−1)+1r=6(m-1)+1r=6(m−1)+1 recorders per vertex, R=mrR=mrR=mr recorders total, O(m2)O(m^2)O(m2) qubits, and threshold k=R+m−1k=R+m-1k=R+m−1. Tracking intersections of qubit symplectic planes with a Lagrangian subspace yields appearance-count bounds that prevent single-qubit Clifford shortcuts: any circuit meeting the threshold induces a CNOT-only implementation encoding a Hamiltonian path. Further counting handles output permutations and hence free SWAPs and arbitrary two-qubit Cliffords. Polynomial-size certificates establish membership in NP.
The results concern exact gate-count optimization without extra ancillas, not depth optimization. They establish worst-case decision complexity, rather than approximation hardness or typical-instance difficulty.
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.
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.
Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
To achieve large-scale fault-tolerant quantum computation, it may be easier to combine many small sets of qubits than to construct a single large set. For example via quantum error correction with block codes, or distributed quantum processors utilizing shared entanglement. In these regimes, the time or error budget of the overall quantum computation may be dominated by non-local operations. Hence, it is worthwhile to minimize the number of these operations. We consider the case where both non-local and local connectivity may be arbitrarily restricted, and give an asymptotically optimal synthesis method for distributed CNOT and Clifford circuits, based on block-matrix Gaussian elimination. We extend this to all Clifford+RZ circuits by generalizing the Pauli exponential circuit representation; this naturally integrates with existing methods for optimizing T-count. As an application, we show how to implement CNOT circuits in a CSS code encoding n logical qubits in k blocks using O(nk) inter-block transversal CNOTs and intra-block Pauli measurements.
Exact CNOT synthesis seeks a minimum-size circuit implementing a given invertible binary linear transformation. We study its vanilla formulation: synthesis from the identity on fixed labeled qubits, without ancillas and with all-to-all connectivity. We prove that this problem is NP-hard and that its bounded decision version is NP-complete. Hardness persists even for unitriangular targets that are low-rank perturbations of the identity and under near-linear gate budgets. Our proof gives a polynomial-time reduction from the NP-complete Hamiltonian path problem on grid graphs. A unary hypercube embedding translates graph vertices into circuit parities and edge traversals into CNOT updates. The central challenge is that Hamiltonian paths require intermediate vertex visits, whereas exact synthesis constrains only the final transformation. We bridge this gap with a replicated recorder qubits construction that encodes the required parities in the final outputs, together with a tight gate budget that forces every sufficiently short implementation to trace a Hamiltonian path. Crucially, this path structure is enforced by the construction rather than imposed as a restriction on the circuit. The result directly implies NP-hardness for shortest word problem and Cayley-graph distance computation over GL(n,2)\mathrm{GL}(n,2)GL(n,2) with elementary transvections as generators, as well as fixed-width sequential XOR program minimization. For CNOT-minimal exact phase polynomial synthesis, it yields NP-hardness with a zero phase polynomial; extending the reduction to grid-graph Hamiltonian cycle problem establishes hardness even when the final linear transformation is the identity. These complementary results show that the linear and phase components each independently suffice for NP-hardness of exact phase polynomial synthesis.
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.