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arxiv: 2603.25101 · v2 · pith:LS6LU542new · submitted 2026-03-26 · 🪐 quant-ph · cs.ET

T Count as a Numerically Solvable Minimization Problem

Marc Grau Davis , Ed Younis , Mathias Weiden , Hyeongrak Choi , Dirk Englund This is my paper

Pith reviewed 2026-05-19 17:24 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET
keywords T-countquantum circuit synthesisnumerical optimizationfault tolerant quantum computingunitary decomposition
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The pith

The minimal T-count for implementing a unitary is found by binary search over continuous minimization problems that prove solvable in practice.

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

The paper recasts the search for the smallest number of T gates required to realize a given quantum unitary as a binary search over a series of continuous minimization problems. Numerical solvers are shown to succeed on these problems, recovering the best known T-counts for small-qubit circuits and extending the reachable size. Partitioning the circuit into independent sub-circuits further scales the method to larger qubit counts. A sympathetic reader cares because T-count directly measures the non-Clifford resources needed in error-corrected quantum computation, so better automated synthesis could lower the overhead of running algorithms.

Core claim

The problem of finding the smallest T-count circuit implementing a unitary is formulated as a binary search over continuous minimization problems, and these problems are demonstrated to be numerically solvable in practice for small to medium sized circuits.

What carries the argument

Binary search over continuous minimization problems for T-count feasibility.

If this is right

  • Best-known T-count results for small qubit counts are reproduced.
  • Bounds on the largest solvable circuits are pushed forward.
  • Circuit partitioning allows independent optimization of sub-circuits for large qubit numbers.

Where Pith is reading between the lines

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

  • Combining this numerical approach with existing algebraic synthesis methods might yield hybrid tools that handle even bigger instances.
  • Applying the method to specific quantum algorithms could reveal lower T-count implementations than hand-optimized ones.
  • The success of continuous relaxations suggests similar formulations may work for other gate-count problems in quantum synthesis.

Load-bearing premise

The continuous relaxation and numerical minimization accurately detect the existence of a discrete T-count circuit without being trapped in local minima that would produce false negatives on feasible counts.

What would settle it

A calculation showing that the numerical method returns a higher than known minimal T-count for a specific unitary where the minimal circuit is established by other means.

Figures

Figures reproduced from arXiv: 2603.25101 by Dirk Englund, Ed Younis, Hyeongrak Choi, Marc Grau Davis, Mathias Weiden.

Figure 1
Figure 1. Figure 1: FIG. 1. Synthesis error cost of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Cost Function Behavior [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We present a formulation of the problem of finding the smallest T -Count circuit that implements a given unitary as a binary search over a sequence of continuous minimization problems, and demonstrate that these problems are numerically solvable in practice. We reproduce best-known results for synthesis of circuits with a small number of qubits, and push the bounds of the largest circuits that can be solved for in this way. Additionally, we show that circuit partitioning can be used to adapt this technique to be used to optimize the T -Count of circuits with large numbers of qubits by breaking the circuit into a series of smaller sub-circuits that can be optimized independently.

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

3 major / 2 minor

Summary. The paper formulates finding the minimal T-count circuit implementing a given unitary as a binary search over a sequence of continuous minimization problems. It claims these problems are numerically solvable in practice, reproduces best-known T-counts for small-qubit cases, extends the approach to larger instances, and introduces circuit partitioning to handle high-qubit circuits by optimizing sub-circuits independently.

Significance. If the continuous relaxations reliably reach global minima that correspond to valid discrete T-count circuits, the method could offer a scalable numerical tool for T-count optimization in Clifford+T synthesis, extending beyond exact solvers limited to very small n. The reproduction of known optimal results for small systems provides initial evidence of practicality, though this strength is tempered by the absence of supporting convergence diagnostics.

major comments (3)
  1. [Results (reproduction of best-known small-n cases)] The central claim that the problems are 'numerically solvable in practice' and correctly recover known T-counts rests on the continuous relaxation accurately indicating feasibility without false negatives from local minima. However, the results section provides no multi-start statistics, convergence curves, or final cost values across random initializations to substantiate that the reported minima are global.
  2. [Numerical method and small-qubit experiments] No explicit verification is given that the numerical solutions (after minimization) correspond to valid discrete T-count circuits, such as by checking that the obtained parameters round to integer T-gate placements or by reconstructing and validating the circuit unitary. This leaves open whether the binary search may over-estimate the minimal count.
  3. [Circuit partitioning section] The partitioning technique for large-qubit circuits assumes sub-circuit optimizations can be combined without increasing the total T-count at interfaces, but no quantitative example or bound is provided showing that the sum of sub-circuit T-counts equals the global optimum.
minor comments (2)
  1. [Abstract] The abstract states that 'best-known results are reproduced' but does not list the specific circuits or qubit numbers tested, nor quantify any improvements over prior work.
  2. [Methods] Notation for the continuous penalty or relaxation term should be introduced with an explicit equation in the methods section for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our numerical approach to T-count minimization. We address each major comment in turn below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Results (reproduction of best-known small-n cases)] The central claim that the problems are 'numerically solvable in practice' and correctly recover known T-counts rests on the continuous relaxation accurately indicating feasibility without false negatives from local minima. However, the results section provides no multi-start statistics, convergence curves, or final cost values across random initializations to substantiate that the reported minima are global.

    Authors: We agree that reporting additional convergence diagnostics would provide stronger evidence that the observed minima are global. In the revised manuscript we will add multi-start statistics (e.g., success rate over 50 random initializations), representative convergence curves, and tables of final cost values for the small-qubit benchmark cases. These additions will quantify the reliability of the numerical solver and directly address the concern about local minima. revision: yes

  2. Referee: [Numerical method and small-qubit experiments] No explicit verification is given that the numerical solutions (after minimization) correspond to valid discrete T-count circuits, such as by checking that the obtained parameters round to integer T-gate placements or by reconstructing and validating the circuit unitary. This leaves open whether the binary search may over-estimate the minimal count.

    Authors: We acknowledge that an explicit post-processing verification step is currently only implicit in the reproduction of known results. In the revision we will insert a dedicated subsection describing the rounding procedure applied to the continuous parameters, followed by explicit unitary reconstruction and fidelity checks (within numerical tolerance) for all reported small-qubit instances. This will confirm that the binary search yields valid discrete T-count circuits and rule out over-estimation. revision: yes

  3. Referee: [Circuit partitioning section] The partitioning technique for large-qubit circuits assumes sub-circuit optimizations can be combined without increasing the total T-count at interfaces, but no quantitative example or bound is provided showing that the sum of sub-circuit T-counts equals the global optimum.

    Authors: The partitioning method is presented as a scalable heuristic rather than a provably optimal procedure. We will revise the relevant section to state this limitation explicitly and will add a concrete quantitative example on a medium-sized circuit (e.g., 6–8 qubits) that compares the partitioned T-count against a known or independently computed reference value, together with a short discussion of possible interface overhead. This will illustrate the practical performance of the heuristic while clarifying its scope. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical formulation and solvability claims are independent of inputs

full rationale

The paper formulates T-count minimization as a binary search over continuous penalty problems and reports that these are solvable in practice via numerical optimization, with validation through reproduction of known small-qubit results and extension via partitioning. No derivation step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the continuous relaxation is presented as an independent modeling choice whose success is checked externally against known benchmarks rather than being forced by the same data. The central claim of practical solvability rests on reported optimizer performance, not on any tautological equivalence to the discrete problem definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the approach implicitly relies on standard numerical optimization assumptions.

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Reference graph

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