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arxiv: 2604.05660 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Coherence and Imaginarity as Resources in Quantum Circuit Complexity

Linlin Ye , Zhaoqi Wu , Nanrun Zhou This is my paper

Pith reviewed 2026-05-10 19:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum circuit complexitycoherenceimaginarityTsallis entropyquantum resourceslower boundsT gate
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The pith

Lower bounds on quantum circuit cost are obtained from coherence and imaginarity using Tsallis relative entropy.

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

The paper shows that measures of coherence and imaginarity can place lower limits on the smallest number of gates needed to implement a given quantum operation. Using the Tsallis relative α entropy applied to the cohering power, the authors derive a bound tighter than an earlier result under certain conditions, along with explicit estimates for standard gates. They further introduce bounds from the imaginaring power of the circuit, which remain positive for gates such as the T gate even when coherence-based bounds are zero. This work connects two resource theories to the practical question of circuit size in quantum computation.

Core claim

The circuit cost admits a lower bound given by the Tsallis relative α entropy of the cohering power that improves on the bound of Bu et al. under the stated restrictive conditions; the same entropy and the ordinary relative entropy applied to imaginaring power supply additional lower bounds. These imaginaring bounds are nontrivial for the T gate, where coherence measures yield a zero lower bound, indicating that imaginarity can sometimes supply stricter constraints than coherence alone.

What carries the argument

Tsallis relative α entropy of cohering power and imaginaring power, which convert the resource-generating ability of a circuit into a numerical lower bound on the number of gates required.

If this is right

  • Explicit lower bounds are obtained for the circuit cost of typical gates including the T gate.
  • Relationships are established between circuit cost and average coherence generating power expressed through skew information and relative entropy.
  • Imaginarity supplies positive lower bounds in cases where all coherence-based bounds vanish.

Where Pith is reading between the lines

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

  • The same resource measures could be applied to estimate the gate cost of larger composite circuits by additivity or subadditivity arguments.
  • Imaginarity bounds might be tested in experimental platforms that realize non-Clifford gates where phase information is costly to generate.
  • The approach suggests that other resource theories could be combined with coherence to refine complexity estimates for fault-tolerant gate sets.

Load-bearing premise

The inequalities relating circuit cost to these entropy measures hold only when the circuit satisfies the restrictive conditions given in the paper.

What would settle it

An explicit quantum circuit whose minimal gate count falls below the numerical value of the Tsallis relative α entropy of its cohering power would disprove the claimed lower bound.

Figures

Figures reproduced from arXiv: 2604.05660 by Linlin Ye, Nanrun Zhou, Zhaoqi Wu.

Figure 1
Figure 1. Figure 1: Cost(U) is the length of the shortest admissible path from I to U on SU(d n ). Denote by H a d dimensional Hilbert space, and D(H) the set of all density operators on H. The Tsallis relative α entropy provides a one-parameter generalization of the quantum relative entropy, which is given by [76, 77] Dα(ρkσ) = 1 α − 1 [fα(ρ, σ) − 1] , α ∈ (0, 1) ∪ (1, +∞), (3) where fα(ρ, σ) = Tr ρ ασ 1−α  . When α → 1, th… view at source ↗
Figure 2
Figure 2. Figure 2: The lower bounds of Cost(U). (a) U = Uθ, where the lower bound is denoted by f(θ) = max  1 16 1 − 3π 16  sin2 (2θ), sin4 θ ln(sin2 θ)−cos4 θ ln(cos2 θ) 8 ln 2 cos 2θ  ; (b) U = Ut , where the lower bound is denoted by g(t) = t 2 ln t−(1−t) 2 ln(1−t) 16 ln 2(1−2t) . For the Grover iteration in Grover’s search algorithm, we obtain CGPS(G) ≈ 0.0654 and CGPR(G) ≈ 0.1487, which in turn imply the lower bound … view at source ↗
read the original abstract

Quantum circuit complexity quantifies the minimal number of gates needed to realize a unitary transformation and plays a central role in quantum computation. In this work, we investigate the complexity of quantum circuits through coherence and imaginarity resources. We establish a lower bound on the circuit cost by the Tsallis relative $\alpha$ entropy of cohering power, which is shown to be tighter than the one presented by Bu et al.[\textit{Communications in Mathematical Physics} 405, no. 7 (2024):161] under restrictive conditions. As a consequence, we obtain the relationships between the circuit cost and the coherence generating power via probabilistic average in terms of skew information/relative entropy, and present explicit bounds of the circuit cost for typical quantum gates. Moreover, we derive lower bounds on the circuit cost via the imaginaring power of the circuit, induced by the Tsallis relative $\alpha$ entropy and relative entropy. We demonstrate that imaginarity can yield nontrivial constraints on the circuit cost even when coherence-based lower bounds are zero (e.g., for the $T$ gate), which implies that imaginarity may provide advantages under certain circumstances compared with coherence. Our results may help better understand the connections between quantum resources and circuit complexity.

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 / 2 minor

Summary. The paper claims to establish lower bounds on quantum circuit cost using the Tsallis relative α-entropy of cohering power (tighter than Bu et al. under restrictive conditions), derive relations to coherence generating power via skew information and relative entropy, provide explicit bounds for typical gates, and obtain lower bounds from imaginarity power. It highlights that imaginarity yields nontrivial constraints for the T gate even when coherence-based bounds vanish.

Significance. If the monotonicity and chaining properties hold under the stated conditions, the work strengthens links between resource theories and circuit complexity by offering potentially tighter bounds and demonstrating imaginarity's complementary role. The explicit gate examples and comparison to prior results provide concrete illustrations that could inform complexity analyses, though the restrictiveness noted in the abstract limits broad applicability.

major comments (2)
  1. [Section on main bound (likely §3 or Theorem 1)] The central lower bound via Tsallis relative α-entropy of cohering power (claimed tighter than Bu et al.) is derived only under restrictive conditions whose precise statement and verification for standard universal gate sets (e.g., Clifford+T) are load-bearing; if monotonicity of the cohering power under the allowed gates or the required subadditivity across circuit layers fails for the chosen α, both the tightness claim and the T-gate example collapse. The manuscript should include an explicit check that these conditions are satisfied without post-hoc restrictions that exclude common non-Clifford circuits.
  2. [Section deriving imaginarity bounds and T-gate example] The imaginarity-based lower bound for the T gate (where coherence vanishes) requires that the imaginarity power induced by Tsallis relative α-entropy and relative entropy is monotonic under the circuit's allowed operations and satisfies the chaining property; without a direct verification in the derivation that this holds for the T gate itself, the claim that imaginarity provides an advantage remains conditional on the same restrictive assumptions.
minor comments (2)
  1. [Preliminaries or definitions section] Clarify the definition of 'cohering power' and 'imaginarity power' at first use, including how they are computed from the Tsallis relative entropy; this would aid readability when relating to skew information.
  2. [Introduction and main results] The abstract and introduction note the restrictiveness of conditions; move a concise statement of these conditions to the main theorem statement for immediate visibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to incorporate explicit verifications as suggested.

read point-by-point responses

  1. Referee: [Section on main bound (likely §3 or Theorem 1)] The central lower bound via Tsallis relative α-entropy of cohering power (claimed tighter than Bu et al.) is derived only under restrictive conditions whose precise statement and verification for standard universal gate sets (e.g., Clifford+T) are load-bearing; if monotonicity of the cohering power under the allowed gates or the required subadditivity across circuit layers fails for the chosen α, both the tightness claim and the T-gate example collapse. The manuscript should include an explicit check that these conditions are satisfied without post-hoc restrictions that exclude common non-Clifford circuits.

    Authors: The restrictive conditions required for monotonicity and subadditivity of the Tsallis relative α-entropy of cohering power are explicitly stated in the assumptions preceding Theorem 1 and in Section 3. These conditions are necessary for the bound to hold and are not introduced post hoc. For the Clifford+T gate set, the T-gate example satisfies the conditions for the chosen α, as the cohering power remains well-defined and the chaining property applies under circuit composition. To strengthen the presentation and directly address the concern, we will add an explicit verification subsection (or appendix) confirming that the properties hold for standard universal gate sets, including Clifford+T, without excluding common non-Clifford circuits. revision: yes

  2. Referee: [Section deriving imaginarity bounds and T-gate example] The imaginarity-based lower bound for the T gate (where coherence vanishes) requires that the imaginarity power induced by Tsallis relative α-entropy and relative entropy is monotonic under the circuit's allowed operations and satisfies the chaining property; without a direct verification in the derivation that this holds for the T gate itself, the claim that imaginarity provides an advantage remains conditional on the same restrictive assumptions.

    Authors: The imaginarity bounds in Section 4 are derived under the same conditions as the coherence case, ensuring monotonicity and the chaining property for the Tsallis relative α-entropy and relative entropy. The T gate is selected because it induces nonzero imaginarity power while the coherence-based bound vanishes, and the properties hold for this gate within the stated framework. We agree that a direct verification would remove any ambiguity. We will therefore insert a concise verification paragraph in the T-gate example subsection demonstrating that monotonicity and chaining are satisfied specifically for the T gate, rendering the advantage claim explicit rather than conditional. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from independent resource measures

full rationale

The paper defines cohering power and imaginaring power via standard Tsallis relative α-entropy and relative entropy of coherence/imaginarity, which are independent of circuit cost. Lower bounds on circuit cost follow from monotonicity and subadditivity properties of these measures under gates, with explicit comparisons to Bu et al. under stated restrictive conditions and direct calculations for gates like T. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The results are self-contained against external benchmarks for resource theory.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definitions of quantum coherence and imaginarity in resource theory, the monotonicity properties of Tsallis relative entropy, and the operational definition of circuit cost as minimal gate count. No new entities are postulated.

free parameters (1)
  • α (Tsallis parameter)
    Continuous parameter in the generalized entropy; its value is chosen to obtain the stated bounds but is not fitted to data in the abstract.
axioms (2)
  • standard math Monotonicity and convexity properties of Tsallis relative α-entropy under quantum channels
    Invoked to derive lower bounds from resource measures.
  • domain assumption Circuit cost is defined as the minimal number of elementary gates realizing the unitary
    Standard definition in quantum circuit complexity.

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