Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.19732

Measurement-Assisted Clifford Synthesis

Sowmitra Das

quant-phSubmitted Nov 24, 2025• Announced Nov 26, 2025• 0 citations
arXivHTMLPDF

Abstract

In this letter, we introduce a method to synthesize an $n$-qubit Clifford unitary $C$ from the stabilizer tableau of its inverse $C†$, using ancilla qubits and measurements. The procedure uses ancillary $|+\rangle$ states, controlled-Paulis, $X$-basis measurements and single-qubit Pauli corrections on the data qubits (based on the measurement results). This introduces a new normal form for Clifford synthesis, with the number of two-qubit gates required exactly equal to the weight of the stabilizer tableau, and a depth linear in $n$.

Related papers

Synthesis of unitaries with Clifford+T circuits

Vadym Kliuchnikov

We describe a new method for approximating an arbitrary $n$ qubit unitary with precision $\varepsilon$ using a Clifford and T circuit with $O(4^{n}n(\log(1/\varepsilon)+n))$ gates. The method is based on rounding off a unitary to a unitary over the ring $\mathbb{Z}[i,1/\sqrt{2}]$ and employing exact synthesis. We also show that any $n$ qubit unitary over the ring $\mathbb{Z}[i,1/\sqrt{2}]$ with entries of the form $(a+b\sqrt{2}+ic+id\sqrt{2})/2^{k}$ can be exactly synthesized using $O(4^{n}nk)$ Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring $O(3^{2^{n}}nk)$ elementary gates.

arXiv:1306.3200•Jun 13, 2013•24 citations

Synthesizing Circuits from Tableaus

anne mara

Quantum computers are only as powerful as the compilers that optimize the circuits they run. As quantum hardware scales, reducing gate counts and circuit depth becomes critical for practical computation. This work implements a core synthesis algorithm in OCaml that maps arbitrary Cliffords to circuits, a fundamental building block in quantum circuit compilation. The Clifford synthesis implementation is designed around how LambdaPC thinks about quantum computation, through Pauli operators. It frames synthesis as a search problem: given a Clifford tableau, find a path to the identity by applying gates. Working factor by factor, it reduces each Pauli pair (P, Q) to a canonical (Z_i, X_i) form using single-qubit Clifford gates and TQE gates, until the full tableau is decomposed into an executable circuit. The main challenge was working within LambdaPC's typed representation, where n-qubit Paulis are encoded as nested pairs. This required helper functions to extract Pauli data from those structures and pattern-matching logic to find minimal gate sequences, laying the groundwork for end-to-end Clifford compilation in LambdaPC.

•Jun 8, 2026•0 citations

Approximate synthesis of general single-qubit unitaries over the Clifford+$\sqrt{T}$ gate set

Mathias Weiden, Jae Won Kim, Justin Kalloor +2 more

For the standard Clifford+$T$ gate set, deterministic, ancilla-free synthesis now attains the minimal $T$-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The $\sqrt{T}$ gate rotates by half the angle of $T$, 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+$\sqrt{T}$ 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 $\varepsilon=10^{-3}$ to $10^{-8}$, the cost of Clifford+$\sqrt{T}$ circuits scales as $2.4\log_2(1/\varepsilon)$ compared to $3.0\log_2(1/\varepsilon)$ for $T$-count-optimal Clifford+$T$ circuits. Once a one-time catalyst state is amortized, the Clifford+$\sqrt{T}$ circuits are never costlier than their Clifford+$T$ counterparts.

quant-pharXiv:2609.16659•Sep 16, 2026•0 citations

A graph-state based synthesis framework for Clifford isometries

Timothée Goubault de Brugière, Simon Martiel, Christophe Vuillot

We tackle the problem of Clifford isometry compilation, i.e, how to synthesize a Clifford isometry into an executable quantum circuit. We propose a simple framework for synthesis that only exploits the elementary properties of the Clifford group and one equation of the symplectic group. We highlight the versatility of our framework by showing that several normal forms of the literature are natural corollaries. We recover the state of the art two-qubit gate depth necessary for the execution of a Clifford circuit on an LNN architecture, concomitantly with another work. We also propose practical synthesis algorithms for Clifford isometries with a focus on Clifford operators, graph states and codiagonalization of Pauli rotations. Benchmarks show that in all three cases we improve the 2-qubit gate count and depth of random instances compared to the state-of-the-art methods. We also improve the execution of practical quantum chemistry experiments.

arXiv:2212.06928•Quantum•Jan 14, 2025•4 citations

Optimal ancilla-free Clifford+T synthesis for general single-qubit unitaries

Hayata Morisaki, Sano, Kaoru, Seiseki Akibue

We propose two Clifford+$T$ synthesis algorithms that are optimal with respect to $T$-count. The first algorithm, called deterministic synthesis, approximates any single-qubit unitary by a single-qubit Clifford+$T$ circuit with the minimum $T$-count. The second algorithm, called probabilistic synthesis, approximates any single-qubit unitary by a probabilistic mixture of single-qubit Clifford+$T$ circuits with the minimum $T$-count. For most of single-qubit unitaries, the runtimes of deterministic synthesis and probabilistic synthesis are $\varepsilon^{-1/2 - o(1)}$ and $\varepsilon^{-1/4 - o(1)}$, respectively, for an approximation error $\varepsilon$. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at $\varepsilon \approx 10^{-15}$ and $\varepsilon \approx 10^{-22}$, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most $3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates, and the probabilistic synthesis algorithm requires at most $1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.

arXiv:2510.05816•Oct 7, 2025•0 citations

Depth reduction for quantum Clifford circuits through Pauli measurements

Yi-Cong Zheng, Ching–Yi Lai, Todd A. Brun +1 more

Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|Φ^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, $O(1)$ CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by $O(1)$ steps of Steane syndrome extraction if clean CSS stabilizer states are available.

arXiv:1805.12082•May 30, 2018•2 citations
In this letter, we introduce a method to synthesize an nnn-qubit Clifford unitary CCC from the stabilizer tableau of its inverse C†C†C†, using ancilla qubits and measurements. The procedure uses ancillary ∣+⟩|+\rangle∣+⟩ states, controlled-Paulis, XXX-basis measurements and single-qubit Pauli corrections on the data qubits (based on the measurement results). This introduces a new normal form for Clifford synthesis, with the number of two-qubit gates required exactly equal to the weight of the stabilizer tableau, and a depth linear in nnn.
We describe a new method for approximating an arbitrary nnn qubit unitary with precision ε\varepsilonε using a Clifford and T circuit with O(4nn(log⁡(1/ε)+n))O(4^{n}n(\log(1/\varepsilon)+n))O(4nn(log(1/ε)+n)) gates. The method is based on rounding off a unitary to a unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] and employing exact synthesis. We also show that any nnn qubit unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] with entries of the form (a+b2+ic+id2)/2k(a+b\sqrt{2}+ic+id\sqrt{2})/2^{k}(a+b2​+ic+id2​)/2k can be exactly synthesized using O(4nnk)O(4^{n}nk)O(4nnk) Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring O(32nnk)O(3^{2^{n}}nk)O(32nnk) elementary gates.
Quantum computers are only as powerful as the compilers that optimize the circuits they run. As quantum hardware scales, reducing gate counts and circuit depth becomes critical for practical computation. This work implements a core synthesis algorithm in OCaml that maps arbitrary Cliffords to circuits, a fundamental building block in quantum circuit compilation. The Clifford synthesis implementation is designed around how LambdaPC thinks about quantum computation, through Pauli operators. It frames synthesis as a search problem: given a Clifford tableau, find a path to the identity by applying gates. Working factor by factor, it reduces each Pauli pair (P, Q) to a canonical (Z_i, X_i) form using single-qubit Clifford gates and TQE gates, until the full tableau is decomposed into an executable circuit. The main challenge was working within LambdaPC's typed representation, where n-qubit Paulis are encoded as nested pairs. This required helper functions to extract Pauli data from those structures and pattern-matching logic to find minimal gate sequences, laying the groundwork for end-to-end Clifford compilation in LambdaPC.
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.
We tackle the problem of Clifford isometry compilation, i.e, how to synthesize a Clifford isometry into an executable quantum circuit. We propose a simple framework for synthesis that only exploits the elementary properties of the Clifford group and one equation of the symplectic group. We highlight the versatility of our framework by showing that several normal forms of the literature are natural corollaries. We recover the state of the art two-qubit gate depth necessary for the execution of a Clifford circuit on an LNN architecture, concomitantly with another work. We also propose practical synthesis algorithms for Clifford isometries with a focus on Clifford operators, graph states and codiagonalization of Pauli rotations. Benchmarks show that in all three cases we improve the 2-qubit gate count and depth of random instances compared to the state-of-the-art methods. We also improve the execution of practical quantum chemistry experiments.
We propose two Clifford+TTT synthesis algorithms that are optimal with respect to TTT-count. The first algorithm, called deterministic synthesis, approximates any single-qubit unitary by a single-qubit Clifford+TTT circuit with the minimum TTT-count. The second algorithm, called probabilistic synthesis, approximates any single-qubit unitary by a probabilistic mixture of single-qubit Clifford+TTT circuits with the minimum TTT-count. For most of single-qubit unitaries, the runtimes of deterministic synthesis and probabilistic synthesis are ε−1/2−o(1)\varepsilon^{-1/2 - o(1)}ε−1/2−o(1) and ε−1/4−o(1)\varepsilon^{-1/4 - o(1)}ε−1/4−o(1), respectively, for an approximation error ε\varepsilonε. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at ε≈10−15\varepsilon \approx 10^{-15}ε≈10−15 and ε≈10−22\varepsilon \approx 10^{-22}ε≈10−22, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most 3log⁡2(1/ε)+o(log⁡2(1/ε))3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))3log2​(1/ε)+o(log2​(1/ε)) TTT-gates, and the probabilistic synthesis algorithm requires at most 1.5log⁡2(1/ε)+o(log⁡2(1/ε))1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))1.5log2​(1/ε)+o(log2​(1/ε)) TTT-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.
Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an nnn-qubit Clifford circuit UUU can be implemented by preparing an ancillary stabilizer state (I⊗U)∣Φ+⟩⊗n(I\otimes U)|Φ^+\rangle^{\otimes n}(I⊗U)∣Φ+⟩⊗n for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing O(1)O(1)O(1) ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, O(1)O(1)O(1) CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by O(1)O(1)O(1) steps of Steane syndrome extraction if clean CSS stabilizer states are available.