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arxiv: 2604.20033 · v1 · submitted 2026-04-21 · 🪐 quant-ph

Direct U(2) approximation via repeat-until-success circuits

Vadym Kliuchnikov , Jendrik Brachter , Marcus P. da Silva This is my paper

Pith reviewed 2026-05-10 01:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum circuitsunitary approximationrepeat-until-successsingle-qubit gatesancilla qubitClifford gateslattice synthesis
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The pith

Repeat-until-success circuits approximate arbitrary one-qubit unitaries directly with one ancillary qubit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates a technique that approximates any single-qubit unitary operation without first decomposing it into Euler angles or handling magnitude separately. The method relies on repeat-until-success circuits built from lattice-based exact synthesis, integer point enumeration in convex sets, and solutions to relative norm equations. This comes at the modest cost of one extra qubit but removes intermediate approximation steps that often complicate circuit design. The same construction extends to approximating unitaries under multi-qubit gate sets such as Clifford plus CS or Clifford plus CCZ, and to orthogonal matrices under real Clifford plus CCZ gates.

Core claim

Arbitrary one-qubit unitaries in U(2) can be approximated directly by repeat-until-success circuits that are synthesized exactly from lattices and solved via relative norm equations, eliminating the need for Euler decomposition and separate magnitude approximation while using only one ancillary qubit.

What carries the argument

Repeat-until-success circuits constructed via lattice-based exact synthesis algorithms combined with integer point enumeration in convex sets and relative norm equations.

If this is right

  • Approximations of unitaries become possible directly from Clifford and CS gate sets.
  • The same direct method works for Clifford and CCZ gate sets.
  • Orthogonal matrices can be approximated using real Clifford and CCZ gates.
  • Circuit design for single-qubit operations simplifies by removing separate decomposition and scaling stages.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Compilation tools for quantum algorithms could incorporate this as a standard single-qubit block to reduce total gate count.
  • The technique may generalize to higher-dimensional unitaries if similar lattice methods scale.
  • Hardware implementations with limited ancilla qubits could test whether the one-qubit overhead remains acceptable in practice.

Load-bearing premise

That repeat-until-success circuits can be efficiently constructed for arbitrary unitaries using lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.

What would settle it

Finding a one-qubit unitary for which the lattice synthesis and relative norm equation methods fail to produce an efficient repeat-until-success circuit, or for which the resulting approximation requires more resources than Euler-based methods.

Figures

Figures reproduced from arXiv: 2604.20033 by Jendrik Brachter, Marcus P. da Silva, Vadym Kliuchnikov.

Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Repeat-until-success circuit that implements one-qubit unitary [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce a direct method for approximating arbitrary one-qubit unitaries (and extensions to certain multi-qubit gate sets and orthogonal matrices) via repeat-until-success circuits. The approach bypasses Euler decomposition and magnitude approximation by using one ancillary qubit and relies on lattice-based exact synthesis, integer-point enumeration in convex sets, and relative norm equations.

Significance. If the efficiency claims and constructions hold with rigorous bounds, the work would offer a meaningful alternative synthesis technique in quantum compilation, potentially simplifying single-qubit approximation and extending RUS methods to broader gate sets without standard decompositions.

major comments (2)
  1. [Abstract and methods] Abstract and methods: the central claim of an 'efficient' direct approximation for arbitrary targets rests on the polynomial-time constructibility of the RUS circuits via integer point enumeration in convex sets; however, no complexity analysis or scaling with target precision is supplied, and the general #P-hardness of integer-point enumeration in variable-dimension convex bodies is not addressed or circumvented for the arbitrary-U(2) case.
  2. [Abstract and methods] The manuscript asserts that the method bypasses the magnitude approximation problem, but provides no explicit derivation or example showing how the lattice-based construction and relative norm equations achieve this for a general target unitary (as opposed to specially structured cases).
minor comments (1)
  1. [Abstract] The abstract would be strengthened by including a brief statement of the achieved approximation error or the number of RUS iterations required as a function of precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which highlight important aspects of our presentation. We address each major comment point by point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract and methods: the central claim of an 'efficient' direct approximation for arbitrary targets rests on the polynomial-time constructibility of the RUS circuits via integer point enumeration in convex sets; however, no complexity analysis or scaling with target precision is supplied, and the general #P-hardness of integer-point enumeration in variable-dimension convex bodies is not addressed or circumvented for the arbitrary-U(2) case.

    Authors: We acknowledge that the manuscript lacks an explicit complexity analysis. However, the dimension of the relevant convex sets is fixed by the one-qubit U(2) structure and the single-ancilla RUS circuit; it does not increase with the target precision. In fixed dimension, integer-point enumeration in convex bodies is solvable in polynomial time (e.g., via Lenstra's algorithm for integer programming). We will add a dedicated subsection providing this analysis, including the scaling with precision, and explicitly note that the variable-dimension #P-hardness result does not apply. revision: yes

  2. Referee: The manuscript asserts that the method bypasses the magnitude approximation problem, but provides no explicit derivation or example showing how the lattice-based construction and relative norm equations achieve this for a general target unitary (as opposed to specially structured cases).

    Authors: We agree that an explicit derivation and example would improve clarity. The relative norm equations allow the target unitary parameters to be embedded directly into the lattice search, with the RUS success probability incorporated implicitly via the norm conditions, thereby avoiding a separate magnitude-approximation stage. In the revised manuscript we will expand the methods section with a step-by-step derivation for a general U(2) target and include a concrete numerical example. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method applies external synthesis tools

full rationale

The paper's central claim is a construction that directly applies repeat-until-success circuits together with lattice-based exact synthesis, integer-point enumeration, and relative-norm equations to approximate arbitrary U(2) targets. These are presented as pre-existing algorithmic primitives rather than results derived inside the paper. No self-definitional loop, fitted-parameter prediction, or load-bearing self-citation chain appears in the derivation; the technique is framed as a composition of independently established tools that bypasses Euler decomposition. The skeptic concern about efficiency is a question of complexity bounds, not a reduction of the claimed result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; a complete ledger requires the full manuscript for identification of any fitted values or background assumptions.

pith-pipeline@v0.9.0 · 5386 in / 1120 out tokens · 53791 ms · 2026-05-10T01:49:28.063445+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Glaudell, Shaun Kelso, William Maxwell, Samuel S

    Matthew Amy, Andrew N. Glaudell, Shaun Kelso, William Maxwell, Samuel S. Mendelson, and Neil J. Ross. Exact synthesis of multiqubit Clifford-cyclotomic circuits, 2024.arXiv: 2311.07741

    work page arXiv 2024

  2. [2]

    Glaudell, and Neil J

    Matthew Amy, Andrew N. Glaudell, and Neil J. Ross. Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits.Quantum, 4:252, April 2020.doi:10.22331/ q-2020-04-06-252

    work page 2020

  3. [3]

    Quantum Science and Technology , abstract =

    Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. Lower bounds on the non-Clifford resources for quantum computations.Quantum Science and Technology, 5(3):035009, May 2020. URL:http://dx.doi.org/10.1088/2058-9565/ab8963,doi:10. 1088/2058-9565/ab8963

    work page doi:10.1088/2058-9565/ab8963 2020

  4. [4]

    Cambridge University Press, 2004

    Stephen Boyd and Lieven Vandenberghe.Convex Optimization. Cambridge University Press, 2004

    work page 2004

  5. [5]

    Physical Review A 95(4), doi:10.1103/physreva.95.042306

    Earl Campbell. Shorter gate sequences for quantum computing by mixing unitaries.Physical Review A, 95(4), April 2017.doi:10.1103/physreva.95.042306

    work page doi:10.1103/physreva.95.042306 2017

  6. [6]

    Physical Review AAtomic, Molecular, and Optical Physics87(3), 032332 (2013).https: //doi.org/10.1103/PhysRevA.87.032332

    Brett Giles and Peter Selinger. Exact synthesis of multiqubit Clifford circuits.Physical Review A, 87(3), March 2013.doi:10.1103/physreva.87.032332

    work page doi:10.1103/physreva.87.032332 2013

  7. [7]

    Reducing T Gates with Unitary Synthesis,

    Tianyi Hao, Amanda Xu, and Swamit Tannu. Reducing T gates with unitary synthesis. arXiv preprint arXiv:2503.15843, 2025

    work page arXiv 2025

  8. [8]

    Shorter quantum circuits via single-qubit gate approximation , volume=

    Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit. Shorter quantum circuits via single-qubit gate approximation.Quantum, 7:1208, December 2023.doi:10.22331/q-2023-12-18-1208

    work page doi:10.22331/q-2023-12-18-1208 2023

  9. [9]

    Multi-qubit circuit synthesis and Hermi- tian lattices, 2024.arXiv:2405.19302

    Vadym Kliuchnikov and Sebastian Sch¨ onnenbeck. Multi-qubit circuit synthesis and Hermi- tian lattices, 2024.arXiv:2405.19302

    work page arXiv 2024

  10. [10]

    Optimal ancilla-free clifford+ t synthesis for general single-qubit unitaries.arXiv preprint arXiv:2510.05816, 2025

    Hayata Morisaki, Kaoru Sano, and Seiseki Akibue. Optimal ancilla-free Clifford+T synthe- sis for general single-qubit unitaries, 2025.arXiv:2510.05816

    work page arXiv 2025

  11. [11]

    Adam Paetznick and Krysta M. Svore. Repeat-until-success: non-deterministic decompo- sition of single-qubit unitaries.Quantum Info. Comput., 14(15–16):1277–1301, November 2014

    work page 2014

  12. [12]

    Finding the four squares in Lagrange’s theorem.Integers, 18A:A15, 2018

    Paul Pollack and Enrique Trevi˜ no. Finding the four squares in Lagrange’s theorem.Integers, 18A:A15, 2018. 9

    work page 2018