Complexity of detecting large coefficients in the Pauli basis

Abstract: We study the problem of deciding, given a mechanism to prepare a quantum state $ρ$ and a value $\varepsilon > 0$, whether there is some non-identity Pauli matrix $P$ such that $|Tr(P ρ)| \geq \varepsilon$. We consider that the state $ρ$ is described as the result of tracing out some of the qubits of a pure state prepared by a circuit $C$, and we assume the promise that either there is a Pauli matrix satisfying the stated condition or, instead, that for all non-identity Pauli matrices $P$ it is the case that $|Tr(Pρ)|\leq \varepsilon/2$. The problem is in $QCMA$, and we prove that if it belongs to $BQP$ then $NP \subseteq BQP$. The result is obtained through a reduction from the minimum-weight code problem, and it holds even when $ρ$ is assumed to be a pure state (i.e. when no qubits are discarded) and $\varepsilon$ is constant. This resolves an open question regarding the existence of efficient tomographic procedures to find the largest coefficients of a quantum state in the Pauli basis: namely, they do not exist under the standard hypothesis $NP \nsubseteq BQP$.

• The paper establishes that efficiently detecting large Pauli coefficients is BQP-hard, challenging the feasibility of generic selective quantum tomography.
• The methodology uses a reduction from the gap minimum-weight codeword problem to encode Hamming weights, linking quantum complexity with classical coding theory.
• The study implies that practical quantum algorithms must exploit specific state structures, such as Pauli sparsity and low entanglement, for scalable tomography.

Complexity-Theoretic Analysis of Detecting Large Pauli Coefficients

Problem Statement and Motivation

The study addresses the computational complexity of detecting large coefficients in the Pauli basis of quantum states—specifically, given a circuit-generated state $\rho$ and a threshold $\epsilon$, the challenge is to decide if any non-identity Pauli observable $P$ has an expectation value $|\mathrm{Tr}(P\rho)| \geq \epsilon$. The input model considers states produced via quantum circuits, potentially traced over ancillary registers, and formalizes the task as the "Large Pauli Coefficient" (LPC) promise problem. This problem is central to selective and partial quantum tomography, where efficiently identifying significant elements in the Pauli expansion is essential to scalable quantum state characterization.

Complexity-Theoretic Results

A reduction from the $\lambda$-gap minimum-weight codeword (GapMWC) problem establishes the main hardness result. Specifically, a deterministic polynomial-time reduction constructs a quantum state whose relevant Pauli expectation values encode the Hamming weights of codewords in a binary linear code. The reduction leverages the identity:

$$\sum_{j=1}^m (-1)^{x \cdot g_j} = m - 2\,\mathrm{wt}(xG)$$

where $x$ is a binary vector and $G$ the generator matrix.

The paper proves that if LPC can be solved in BQP for a constant threshold $\epsilon$, then $\mathrm{NP} \subseteq \mathrm{BQP}$, a widely believed unlikely collapse. This resolves the open question on the existence of efficient selective Pauli tomography for arbitrary quantum states—under standard complexity assumptions ($\epsilon$0), such tomography is infeasible for generic states. Furthermore, the reduction holds for both mixed and pure states via a unitary embedding technique that prevents spurious large coefficients arising from ancillary encodings.

LPC is shown to be in QCMA, as a classical witness (specifying the Pauli matrix) can be verified efficiently via repeated Pauli measurements, with a sample complexity $\epsilon$1. Strong separation is demonstrated: LPC is BQP-hard, meaning every problem in BQP reduces to LPC, further indicating that LPC is not in NP unless $\epsilon$2.

Implications for Quantum Tomography and Practical Quantum Algorithms

The formal results clarify limits on resource-efficient quantum tomography, particularly selective schemes for extracting significant Pauli coefficients. Namely, generic algorithms for detecting large coefficients are not feasible with polynomial quantum and classical resources unless the state exhibits special structure—examples include Pauli sparsity, stabilizer-like structure, locality, low entanglement, or low rank [11, 8, 19, 18, 10].

Practically, the results impact quantum process estimation, shadow tomography, and distributed inner-product estimation [3, 14, 1], as the impossibility result guides algorithm designers towards exploiting state assumptions. It also justifies the exponential-time worst-case complexity in recent hierarchical algorithms for identifying large coefficients [7]. Moreover, the BQP-hardness suggests that no efficient classical witness characterization exists for the problem, barring a collapse of $\epsilon$3 classes.

Theoretical Consequences and Future Directions

The reduction from codeword weight estimation (NP-hard) to observable detection in quantum systems strengthens the connection between quantum computational complexity and classical coding theory. While LPC is in QCMA and hard for NP, a tight inclusion is not established; it is plausible LPC is QCMA-hard, which would further demarcate the boundary of efficient quantum verification.

Algorithmic progress for selective tomography will require leveraging additional structural assumptions—this directs future research towards learning and estimation in structured families (stabilizers, low-rank states, states with few non-Clifford gates [10]). The result underscores the importance of structural model selection in quantum information processing and benchmark applications where scalable state tomography is required.

Conclusion

The paper rigorously establishes the QCMA membership and BQP-hardness of detecting large Pauli coefficients in circuit-prepared quantum states, by reduction from gap minimum-weight code problems. It demonstrates that efficient detection is infeasible for generic states unless $\epsilon$4. The implications for quantum tomography, verification, and state learning are substantial: algorithms must exploit special structure, as generic scalability is precluded by computational complexity. Future progress in selective Pauli tomography will depend on advances in structured quantum state characterization and complexity-theoretic analysis.