In this work we explore a correspondence between quantum circuits and low-degree polynomials over the finite field F_2. Any quantum circuit made up of Hadamard, Z, controlled-Z and controlled-controlled-Z gates gives rise to a degree-3 polynomial over F_2 such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial. We exploit this connection, which is especially clean and simple for this particular gate set, in two directions. First, we give proofs of classical hardness results based on quantum circuit concepts. Second, we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials.
In this work we explore a correspondence between quantum circuits and low-degree polynomials over the finite field F_2. Any quantum circuit made up of Hadamard, Z, controlled-Z and controlled-controlled-Z gates gives rise to a degree-3 polynomial over F_2 such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial. We exploit this connection, which is especially clean and simple for this particular gate set, in two directions. First, we give proofs of classical hardness results based on quantum circuit concepts. Second, we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials.
Abstract In this work we explore a correspondence between quantum circuits and low-degree polynomials over the finite field F 2 . Any quantum circuit made up of Hadamard, Z, controlled-Z and controlled-controlled-Z gates gives rise to a degree-3 polynomial over F 2 such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial. We exploit this connection, which is especially clean and simple for this particular gate set, in two directions. First, we give proofs of classical hardness results based on quantum circuit concepts. Second, we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials.
Montanaro associated to every quantum circuit over the gate set {H,Z,CZ,CCZ} a degree-three polynomial over 𝔽₂, and showed that the strong simulation cost of the circuit is governed by the minimum number of qubit wires needed to realize that polynomial. We study this minimum-width problem. We prove that deciding whether a degree-three polynomial has width at most k is NP-complete. The reduction is parsimonious enough to imply that, for every α < 49/48, the corresponding gap version is NP-hard. We also prove the same inapproximability threshold for degree-two polynomials by a twin-copy replacement of the cubic constraints. On the positive side, we give a nondeterministic search algorithm using only O(klog(en/k)) nondeterministic bits, and a deterministic fixed-parameter algorithm running in k^{6k+o(k)}n+O(m) time on an input with n symbols and m monomials.
A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 × 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. Therefore, it is possible that, The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent with one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial f(x)=∑j=1sajxuj∈Fq[x1±1,…,xn±1], let U be its augmented support matrix whose columns are (1,uj) with rank ρ and N(f):=#{x∈(Fq×)n∣f(x)=0} be its torus point count. Our first main result is a quantum algorithm that outputs N(f) satisfying ∣N(f)−N(f)∣≤εqn+s/2−ρ with success probability 1−δ. Provided that ρ and ∥U∥∞ are bounded, the algorithm runs in both classical bit and quantum gate complexity poly(n,s,logq,1/ε,log(1/δ)). It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in logq. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes #P-hard under randomized polynomial-time Turing reductions when the support matrix U varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.