On the role of coherence for quantum computational advantage

TL;DR

The paper addresses the problem of identifying resources that enable quantum computational advantage by introducing path coherence, a computational measure defined via the sum-over-paths formalism: $pc(\mathcal{C}) = \log|\mathcal{S}_{\boldsymbol{a},\boldsymbol{b}}|$. It then derives a classical Monte Carlo estimator for quantum transition amplitudes with runtime $\mathcal{O}(2^{2 pc(\mathcal{C}) - h} \log(1/\delta) \varepsilon^{-2})$, and shows that when the circuit uses Hadamard gates together with generalised classical linear gates and has $h \le 2n + O(\log n)$ Hadamard gates, the path coherence satisfies $pc(\mathcal{C}) \le h/2$, yielding polynomial-time classical estimation. The work extends to qudit circuits and provides a complexity-theoretic interpretation of a transition between classical and quantum computation, illustrating regimes where classical simulation remains feasible and where quantum advantage requires higher coherence. Practically, these results guide which large classes of quantum circuits (including certain quantum machine learning tasks) can be efficiently simulated classically and help delineate the boundary where coherence must be leveraged for universal quantum computation, with potential extensions to noise settings and other coherence-creating gates.

Abstract

Quantifying the resources available to a quantum computer appears to be necessary to separate quantum from classical computation. Among them, entanglement, nonstabilizerness and coherence are arguably of great significance. We introduce path coherence as a measure of the coherent paths interferences arising in a quantum computation. Leveraging the sum-over-paths formalism, we obtain a classical algorithm for estimating quantum transition amplitudes, the complexity of which scales with path coherence. As path coherence relates to the hardness of classical estimation of quantum transition amplitudes, it provides a new perspective on the role of coherence in quantum computational advantage. Beyond their fundamental significance, our results have practical applications for simulating large classes of quantum computations with classical computers.

Figures (3)

• Figure 1: Venn diagram representing the computational power of resource/gate combinations. In this paper we explore the transition depicted by the red arrow from the point of view of classical simulation. Below a resource name between parentheses is the gate producing that resource. The classical simulation is efficient if any one of these three resources is missing. Classical algorithms for estimating quantum transition amplitudes based on the lack of one resource show that they must be abundant enough for quantum computational advantage.
• Figure 2: Grid traversal for the computation of $\left|\mathcal{Q}_{n,\,6}^{-1}\left(3\right)\right|$ with $h=6$ and $r=3$ -- we assume $n$ is sufficiently large. The goal is therefore the point $(6, 3)$. The dark blue arrow is a direct consequence of \ref{['eq:rankAxIncreases']}. In light blue is illustrated an example of admissible path. The full path (which describes the evolution of the rank) can be written $(1, 1, 1, 2, 3, 3)$, where indeed $\mathcal{A} = \{1, 2, 3\}$ and $\mathcal{I}_{6,3} = \{1, 1, 3\}$ The paths of interest are those from $(1, 1)$ to $(6, 3)$. In total, there are $\binom{5}{2} = 10$ paths.
• Figure 3: Numerical plot of the analytical expression for $\mathop{ \mathrm{Pr} \left[\mathop{\mathrm{rk}}\nolimits(A_x^{(h)}) \geqslant \frac{h}{2}\right]}$ using \ref{['eq:shorthandProbability']}, for ${n=1, \cdots, 10}$ and ${h = 0, \cdots, 2n}$ for each $n$. The plot suggests that, in spite of the asymptotic proof, the probability of having a large enough rank converges quickly to unit probability.

Theorems & Definitions (14)

• Theorem 1: Classical estimation of quantum transition amplitudes
• Theorem 2: Path coherence of random quantum circuits with generalised classical linear gates
• Corollary 1
• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Lemma 1
• proof
• Proposition 2