Abstract An essential problem in quantum machine learning is to find quantum-classical separations between learning models. However, rigorous and unconditional separations are lacking for supervised learning. Here we construct a classification problem defined by a noiseless constant depth (i.e., shallow) quantum circuit and rigorously prove that any classical neural network with bounded connectivity requires logarithmic depth to output correctly with a larger-than-exponentially-small probability. This unconditional near-optimal quantum-classical representation power separation originates from the quantum nonlocality property that distinguishes quantum circuits from their classical counterparts. We further characterize the noise regimes for demonstrating such a separation on near-term quantum devices under the depolarization noise model. In addition, for quantum devices with constant noise strength, we prove that no super-polynomial classical-quantum separation exists for any classification task defined by Clifford circuits, independent of the structures of the circuits that specify the learning models.
Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC
0) can solve with near certainty (probability
1−o(1)), but that any bounded fan-in constant-depth classical circuit (NC
0) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC
0 circuits and AC
0[p] circuits for all primes
p≥2, and a conditional separation between noisy QNC
0 circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks -- that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are
⊕L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.
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.
Recent works by Bravyi, Gosset and König (Science 2018), Bene Watts et al. (STOC 2019), Coudron, Stark and Vidick (QIP 2019) and Le Gall (CCC 2019) have shown unconditional separations between the computational powers of shallow (i.e., small-depth) quantum and classical circuits: quantum circuits can solve in constant depth computational problems that require logarithmic depth to solve with classical circuits. Using quantum error correction, Bravyi, Gosset, König and Tomamichel (Nature Physics 2020) further proved that a similar separation still persists even if quantum circuits are subject to local stochastic noise. In this paper, we consider the case where any constant fraction of the qubits (for instance, huge blocks of qubits) may be arbitrarily corrupted at the end of the computation. We make a first step forward towards establishing a quantum advantage even in this extremely challenging setting: we show that there exists a computational problem that can be solved in constant depth by a quantum circuit but such that even solving any large subproblem of this problem requires logarithmic depth with bounded fan-in classical circuits. This gives another compelling evidence of the computational power of quantum shallow circuits. In order to show our result, we consider the Graph State Sampling problem (which was also used in prior works) on expander graphs. We exploit the "robustness" of expander graphs against vertex corruption to show that a subproblem hard for small-depth classical circuits can still be extracted from the output of the corrupted quantum circuit.
Recent works by Bravyi, Gosset and Konig (Science 2018), Bene Watts et al. (STOC 2019), Coudron, Stark and Vidick (QIP 2019) and Le Gall (CCC 2019) have shown unconditional separations between the computational powers of shallow (i.e., small-depth) quantum and classical circuits: quantum circuits can solve in constant depth computational problems that require logarithmic depth to solve with classical circuits. Using quantum error correction, Bravyi, Gosset, Konig and Tomamichel (Nature Physics 2020) further proved that a similar separation still persists even if quantum circuits are subject to local stochastic noise.
We prove that this quantum advantage persists even if the quantum circuits can be subject to arbitrary corruption: in this paper we assume that any constant fraction of the qubits (for instance, huge blocks of qubits) may be arbitrarily corrupted at the end of the computation. We show that even in this model, quantum circuits can still solve in constant depth computational problems that require logarithmic depth to solve with bounded fan-in classical circuits. This gives another compelling evidence of the computational power of quantum shallow circuits.
In order to show our result, we consider the Graph State Sampling problem (which was also used in prior works) on expander graphs. We exploit the robustness of expander graphs against vertex corruption to show that a subproblem hard for small-depth classical circuits can still be extracted from the output of the corrupted quantum circuit.
Developing classical simulation algorithms for noisy quantum circuits is essential to delineating the limits of quantum advantage. Existing classical sampling approaches for general circuits require circuit depths to grow logarithmically with system size, so that noise drives the global output state close to a trivial state. Here we show that local accumulation of noise at a depth independent of system size is already sufficient. Specifically, we prove that any geometrically local circuit composed of unital operations and interspersed with single-qubit depolarizing noise of strength
p after
Θ(p−1logp−1) depth can be approximately sampled from a polynomial-time classical computer. This generalizes existing results that impose anticoncentration assumptions or non-universal gate sets in order to obtain a classical sampler at such shallow depths. Our proof maps the output state of the circuit to a polymer model in statistical mechanics, and combines a convergent cluster expansion with hypercontractivity of the depolarizing channel.
We consider quantum circuits consisting of
d layers of nearest-neighbor two-qubit gates acting on
n qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error
δ, with parallel runtime
2O(d)loglog(n/δ) and
n2O(d) elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime
O(logn). For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized
AC0-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized
AC0-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth
Ω(logn/loglogn). Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.