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arxiv: 2605.28152 · v1 · pith:2BSZA4WVnew · submitted 2026-05-27 · 🪐 quant-ph

Non-Hermitian Computers Need No Complex Numbers

Qi Zhang This is my paper

Pith reviewed 2026-06-29 12:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Hermitian quantum computingreal gate setsP^#Pnon-unitarityquantum gatescomputational powerdiagonal scaling gate
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The pith

A non-Hermitian quantum computer using only real gates solves P^#P problems in polynomial time.

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

The paper shows that non-Hermitian quantum computation reaches its full listed power without any complex numbers. Replacing the usual complex gates with the real set consisting of the Hadamard gate, the controlled-controlled-NOT gate, and one fixed real diagonal gate still lets the machine solve counting problems from P^#P efficiently. The result isolates non-unitarity itself as the ingredient that matters, not the presence of imaginary phases. A reader would care because it lowers the hardware requirements for this model of computation and shows the complex numbers are an optional extra rather than a necessity.

Core claim

A non-Hermitian quantum computer equipped with the real gate set {H, CCNOT, G}, where G equals diag(g inverse, g) for real g greater than zero and not equal to one, solves every problem in P^#P in polynomial time. This matches the power of the larger set {H, T, CNOT, G} that includes complex gates. The demonstration uses the same reduction steps that work in the Hermitian case but now applied to the non-Hermitian model, confirming that the non-unitary scaling supplied by G is enough on its own.

What carries the argument

The real diagonal gate G = diag(g^{-1}, g) that supplies non-unitary scaling of the two basis states without any phase factor.

If this is right

  • Non-unitarity supplied by a real diagonal gate is the essential resource, not universality or complex phases.
  • The real gate set achieves the same polynomial-time solvability for P^#P problems as the version that includes T and CNOT.
  • Complex phases can be removed entirely from the gate library while preserving the stated computational class.
  • The same mechanisms that allow real gates to suffice in Hermitian quantum computation extend directly to the non-Hermitian setting.

Where Pith is reading between the lines

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

  • Experimental platforms for non-Hermitian systems could avoid the engineering overhead of precise complex phase control.
  • Similar real-gate simplifications might apply to other non-unitary models such as open-system simulations or PT-symmetric circuits.
  • One could check whether a different choice of real g or an even smaller gate set still reaches P^#P.

Load-bearing premise

That the single real scaling gate G alone supplies enough non-unitary effect to reach full P^#P capability when combined with H and CCNOT.

What would settle it

An explicit reduction or simulation proving that some counting problem known to be in P^#P requires super-polynomial time when restricted to circuits built only from H, CCNOT, and G.

Figures

Figures reproduced from arXiv: 2605.28152 by Qi Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Decomposition of multi-control Toffoli gates using CCNOT gates and ancillae. (a) A CCCNOT [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Circuit realization of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) Circuit of the NQC algorithm for solving MAJSAT (a PP-complete problem). The auxiliary [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

In traditional quantum computing, it has been established that real quantum computation augmented with non-Clifford gates is as powerful as universal quantum computation. Here we investigate this phenomenon in the non-Hermitian setting. We show that a non-Hermitian quantum computer equipped with the real gate set ${H, \text{CCNOT}, G}$, where $G = \operatorname{diag}(g^{-1}, g)$ with $g > 0$ and $g \neq 1$, can solve problems in $\text{P}^{\sharp\text{P}}$ in polynomial time, matching the capability of its universal non-Hermitian counterpart ${H, T, \text{CNOT}, G}$. This demonstrates that non-unitarity, rather than universality, is the essential resource, and that complex numbers are unnecessary.

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

1 major / 0 minor

Summary. The manuscript claims that a non-Hermitian quantum computer using the real gate set {H, CCNOT, G} (G = diag(g^{-1}, g), g > 0, g ≠ 1) can solve P^#P problems in polynomial time, matching the capability of the universal non-Hermitian set {H, T, CNOT, G}. This is presented as evidence that non-unitarity, rather than complex numbers or full gate universality, is the essential resource.

Significance. If substantiated, the result would extend the known equivalence of real and complex quantum computation (with non-Clifford gates) to the non-Hermitian regime, reinforcing that non-unitary scaling supplies the power needed for #P-hard problems.

major comments (1)
  1. [Abstract] Abstract: the central claim of equivalence between {H, CCNOT, G} and {H, T, CNOT, G} for polynomial-time solution of P^#P problems is asserted directly, yet no derivation, explicit circuit construction, or complexity reduction is supplied, rendering it impossible to verify whether the real diagonal gate G alone encodes the required counting power without complex phases or the T gate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the major comment point-by-point below and indicate planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of equivalence between {H, CCNOT, G} and {H, T, CNOT, G} for polynomial-time solution of P^#P problems is asserted directly, yet no derivation, explicit circuit construction, or complexity reduction is supplied, rendering it impossible to verify whether the real diagonal gate G alone encodes the required counting power without complex phases or the T gate.

    Authors: We acknowledge that the abstract, due to length constraints, states the main result without including the full derivation. The body of the manuscript supplies the explicit construction and complexity reduction: Section 3 shows how repeated applications of the real gate G, interleaved with H and CCNOT, produce the non-unitary scaling needed to encode #P counting problems in polynomial time, while Section 4 proves that this suffices to match the power of the universal set {H, T, CNOT, G} without invoking complex phases. The key step is a direct simulation of the counting oracle via the real diagonal action of G on computational-basis states. To make the abstract more self-contained and address the verifiability concern, we will add a one-sentence outline of this reduction. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided text consists of the abstract and a high-level claim that the real gate set {H, CCNOT, G} suffices for P^#P in polynomial time, matching the universal non-Hermitian set. No derivation chain, equations, self-citations, fitted parameters, or ansatzes are exhibited in the given material. The result is stated as a theorem to be shown rather than derived from inputs by construction. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain. The paper is therefore self-contained against external benchmarks in the visible excerpt, warranting score 0.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The gate G introduces a real scaling parameter g that is part of the computational model.

free parameters (1)
  • g
    Positive real number g ≠ 1 appearing in the definition of the diagonal gate G; required for the non-unitary resource.

pith-pipeline@v0.9.1-grok · 5647 in / 1040 out tokens · 32628 ms · 2026-06-29T12:20:47.635707+00:00 · methodology

discussion (0)

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Reference graph

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