Comment on Temperature change can solve the Deutsch-Jozsa problem: An exploration of thermodynamic query complexity

Technical Summary: Comment on Temperature change can solve the Deutsch–Jozsa problem: An exploration of thermodynamic query complexity

Problem Statement
This paper addresses a critical flaw in the resource accounting and readout analysis proposed in Phys. Rev. A 113, 012420 (2026) (hereafter Ref. [1]). Ref. [1] claims that the Deutsch–Jozsa decision problem can be solved using a single thermal query (a heat exchange between a probe qubit and a thermal-machine oracle) followed by a number of statistically useful probe samples that is independent of the input size. The authors of this comment argue that the proposed readout mechanism fails to generate independent samples from a single query and that the derived sample lower bounds are mathematically incorrect.

Methodology and Analysis
The paper analyzes the two primary readout strategies discussed in Ref. [1] and the statistical inequalities used to derive sample complexity bounds:

1. CNOT Fanout Analysis: Ref. [1] proposes using CNOT gates to "unitarily reconstruct" a single post-query probe state $\rho_h$ (where $h \in \{B, C\}$ represents balanced or constant hypotheses) into $M$ pure ancilla qubits to allow repeated measurements.

   • The authors demonstrate that applying CNOT gates from a diagonal probe state to $M$ ancillas results in a joint state $\sigma^{(M)}_h = p_h|0^{M+1}\rangle\langle 0^{M+1}| + (1-p_h)|1^{M+1}\rangle\langle 1^{M+1}|$.
   • While individual registers retain the marginal state $\rho_h$, the joint state consists of perfectly correlated registers (either all $0$s or all $1$s).
   • Consequently, the $M+1$ recorded bits constitute a single Bernoulli draw copied $M$ times, not $M+1$ independent draws. The empirical frequency does not converge to the true probability $p_h$ as $M$ increases.
   • The paper calculates the trace distance and relative entropy between the balanced and constant hypotheses for this fanout state, showing they remain identical to the single-copy values ($T(\sigma^{(M)}_B, \sigma^{(M)}_C) = T(\rho_B, \rho_C)$ and $D(\sigma^{(M)}_B \| \sigma^{(M)}_C) = D(\rho_B \| \rho_C)$). Thus, the fanout does not amplify distinguishability.

2. Reset-and-Repeat Analysis: The alternative strategy involves measuring the probe, resetting the system, and repeating the experiment.

   • The authors argue that under the definition in Ref. [1], where a "query" is defined as a heat exchange between the probe and the oracle, this procedure requires $M$ distinct thermal queries to generate $M$ independent samples.
   • Therefore, the claim of "one query followed by many samples" is operationally invalid; independent samples necessitate repeated heat exchanges.

3. Statistical Inequality Critique: The paper examines the sample lower bound $n^*$ stated in Eq. (7) of Ref. [1], which claims $n^* > \ln(1/\delta) / D(\rho_B \| \rho_C)$ and utilizes Pinsker's inequality to relate this to a trace distance threshold $t$.

   • The authors show that Pinsker's inequality ($D \geq 2T^2$) implies an upper bound on the reciprocal of the relative entropy, not a lower bound.
   • Specifically, substituting the inequality yields $\frac{\ln(1/\delta)}{D} < \frac{\ln(1/\delta)}{2t^2}$. A lower bound on $n^*$ derived from the left side cannot logically imply the inequality $n^* > \frac{\ln(1/\delta)}{2t^2}$ stated in Ref. [1].
   • Furthermore, the authors note that the Chernoff–Stein lemma, which governs asymptotic error exponents, does not provide the exact finite-sample inequality used in Ref. [1] without additional analysis.

Key Contributions and Results

• Correction of Resource Counting: The paper establishes that a CNOT fanout of a diagonal post-query probe creates perfectly correlated registers rather than independent copies. Therefore, repeated measurements of these ancillas provide only one Bernoulli observation, not $M$ independent observations.
• Clarification of Query Complexity: The analysis confirms that generating $M$ independent samples via the "reset-and-repeat" method requires $M$ thermal queries, contradicting the claim that the protocol achieves sample independence from input size with a single query.
• Mathematical Refutation of Sample Bounds: The paper demonstrates that the specific sample lower bound cited in Ref. [1] (requiring approximately 116 samples for $\delta = t = 0.1$) does not follow from Pinsker's inequality. The inequality actually provides an upper estimate for the expression, not a necessary lower bound.
• Distinguishability Preservation: It is shown that the CNOT fanout is an isometric embedding that preserves, rather than amplifies, the trace distance and relative entropy between the hypotheses.

Significance and Claims
The authors state that these observations do not invalidate the underlying population-transfer calculations or the possibility that "thermal kickback" encodes decision information in probe temperatures. They also do not rule out the possibility of constant query complexity for a fixed target error, provided the population separation remains bounded away from zero.

However, the paper asserts that the operational conclusions of Ref. [1] must be corrected. Specifically:

1. The claim that a single queried mixed probe yields multiple independent observations is false.
2. Producing the independent observations necessary for temperature estimation entails repeated uses of the heat-exchange oracle.
3. The comparison based on the separation of "one query" versus "many samples" is flawed.
4. The specific numerical claim of "at least 116 samples" derived from Eq. (7) is unsupported by the cited inequalities.

The paper concludes that the one-query readout claim and the associated resource accounting in Ref. [1] require correction to accurately reflect the thermodynamic query complexity.