On the Complexity of the Circuit Width Problem

2026-06-16

AI Summary

This work investigates the computational complexity of determining the quantum circuit width \( w(f) \)—the minimum number of qubits required to implement a given (constant-free) polynomial using H, Z, CZ, and CCZ gates—and establishes that deciding whether \( w(f) \leq k \) is NP-complete. By employing algebraic representations, reduction arguments, and a novel double-copy construction, the authors resolve an open problem posed by Montanaro and extend the hardness result to quadratic polynomials and approximation within a factor of \( 49/48 - \varepsilon \). Additionally, they present a nondeterministic algorithm with witness length \( O(k \log(en/k)) \) and a fixed-parameter algorithm running in time \( k^{6k + o(k)} n + O(m) \).

Abstract

Montanaro's polynomial representation expresses amplitudes of quantum circuits over the gates $H$, $Z$, $CZ$, and $CCZ$ as normalized gaps of degree-three polynomials over $\mathbb{F}_2$. The normalization is governed by the circuit width $w(f)$, the minimum number of qubits in any circuit realizing a polynomial $f$. Thus, efficient width minimization would give an approximate-counting route toward a combinatorial characterization of $BQP$. We study the computational complexity of this parameter. For degree-three polynomials with no constant term, deciding whether $w(f)\le k$ is $NP$-complete, resolving Montanaro's open question. We also prove $NP$-hardness of approximation within any factor $49/48-ε$, and show via a twin-copy construction that the exact and approximation hardness results also hold for degree-two polynomials. Under the Exponential Time Hypothesis, the exact problem admits no $2^{o(n)}$-time algorithm when $k=Θ(n)$. Complementing these hardness results, we give a nondeterministic polynomial-time search algorithm using $2\log_2\binom{n}{k}=O(k\log(en/k))$ witness bits, and a constructive fixed-parameter algorithm parameterized by $k$ with running time $k^{6k+o(k)}n+O(m)$.

Problem

• circuit width
• computational complexity
• quantum circuits
• polynomial representation
• NP-completeness

Innovation

• circuit width
• NP-completeness
• polynomial representation
• fixed-parameter tractability
• quantum circuit complexity