An unconditional distribution learning advantage with shallow quantum circuits

TL;DR

This work proves an unconditional quantum advantage in the probably approximately correct (PAC) distribution learning framework with shallow quantum circuit hypotheses in a meaningful generative distribution learning problem where constant-depth quantum circuits using one and two qubit gates are superior compared to constant-depth bounded fan-in classical circuits.

Abstract

One of the core challenges of research in quantum computing is concerned with the question whether quantum advantages can be found for near-term quantum circuits that have implications for practical applications. Motivated by this mindset, in this work, we prove an unconditional quantum advantage in the probably approximately correct (PAC) distribution learning framework with shallow quantum circuit hypotheses. We identify a meaningful generative distribution learning problem where constant-depth quantum circuits using one and two qubit gates (QNC^0) are superior compared to constant-depth bounded fan-in classical circuits (NC^0) as a choice for hypothesis classes. We hence prove a PAC distribution learning separation for shallow quantum circuits over shallow classical circuits. We do so by building on recent results by Bene Watts and Parham on unconditional quantum advantages for sampling tasks with shallow circuits, which we technically uplift to a hyperplane learning problem, identifying non-local correlations as the origin of the quantum advantage.

Figures (3)

• Figure 1: a) Constant depth quantum circuits with one- and two-qubit gates are compared with b) constant-depth bounded fan-in classical circuits. c) The $\text{majmod}_{p,s}\xspace(x)$ function acting on the finite field $\mathbbm{F}_p$. In this case, $p=23$ and $s=3$. The points are elements in $\mathbbm{F}_p$ and are blue or orange if $\text{majmod}_{p,s}\xspace(x)=0$ or $1$, respectively.
• Figure 2: a) Unitary circuit producing the target distribution $D_{n,p,s}$. The upper box indicates the $n-1$ "edge" qubits of the state vector $\ket{\operatorname{PM}_{n}}$. The lower box indicates the $n$ "vertex" qubits of the same state. Note that the input state vector $\ket{\operatorname{PM}_{n}}$ can be prepared in constant-depth, following \ref{['thm:Binary_Tree_Poor_Man_Construction']}. The gates $U_{m, \frac{\pi}{p}}$ and $C_m$ act on blocks of $m = \left \lceil 2/c + 1 \right \rceil$ qubits, for an arbitrary constant $c\in (0,1/3)$, and are defined in \ref{['appendix:gate_defintions']}. b) Balanced binary tree, where the vertices and edges are assigned to the binary variables $x_1, \ldots, x_{n-1}$ and $d_1, \ldots, d_{n-1}$, respectively, useful for the definition of the state vector $\ket{\operatorname{PM}_{n}}$. c) Visualization of $\cos^2(-\frac{\pi}{4} + \frac{\pi}{p}(k + s))$, a core function in the definition of the target distribution $D_{n,p,s}$. The blue and orange portions of the plot signal the $\text{majmod}_{p,s}\xspace(k) = 0$ and $1$ values, respectively, as in \ref{['fig:majmodp_circle']}.c).
• Figure 3: Constant-depth non-unitary circuit producing an approximate distribution $P$ to $(Z, \text{pmmajmod}_{p,s}\xspace(Z))$, for $\theta = \frac{\pi}{p}$. Note that the input state vector $\ket{\operatorname{PM}_{n}}$ can be prepared in constant-depth, following \ref{['thm:Binary_Tree_Poor_Man_Construction']}.

Theorems & Definitions (18)

• Definition 1: Generator for $D$
• Definition 2: $(\epsilon, \delta)$-PAC generator learner for $\mathcal{D}$
• Theorem 3: Theorem 3 in Ref. WattsParham
• Theorem 4: Theorem 29 in Ref. WattsParham
• Theorem 5: Approximating the desired distribution
• Corollary 6: Optimal quantum circuit
• Theorem 3.1: PAC generator learning algorithm of the quantum distribution
• Theorem 7: Classical hardness of PAC learning
• Theorem 4.1: Advantage of shallow quantum hypotheses
• proof