Quantum Advantage with Noisy Shallow Circuits in 3D

Abstract

Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. It may also be more practical for future implementations. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constantdepth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria. To this end, we provide a protocol for single-shot logical state preparation in the surface code which may be of independent interest. CONTENTS

Quantum unary approach to option pricing

“…Some of the proposed applications include quantum chemistry [4][5][6], simulation of physical systems [7][8][9], combinatorial optimization [10], solving large systems of linear equations [11][12][13], state diagonalization [14,15] or quantum machine learning [16][17][18]. Some exact, non-variational, quantum algorithms are also well suited for NISQ devices [19][20][21][22].…”

“…Figure22. Results for the sampling uncertainties of the expected payoff for increasing number of bins for up to M iterations of Amplitude Estimation for the unary approach, considering depolarizing and read-out errors together.…”

Test of Quantumness with Small-Depth Quantum Circuits

“…Under assumptions such as the non-collapse of the polynomial hierarchy or the hardness of (appropriate versions of) the permanent, strong evidence of the superiority of weak classes of quantum circuits has been obtained from the 2000s [2,3,4,6,12,13,14,18,19,20,33,39]. A recent breakthrough by Bravyi, Gosset and König [10], further strengthened by subsequent works [5,11,17,21], showed an unconditional separation between the computational powers of quantum and classical small-depth circuits by exhibiting a computational task that can be solved by constant-depth quantum circuits but requires logarithmic depth for classical circuits. A major shortcoming, however, is that logarithmic-depth classical computation is a relatively weak complexity class.…”

Constructive nonlocal games with very small classical values

“…Nonlocal games can be regarded as a particular type of Bell-type inequalities and it is well known that Bell-type inequalities [1] are at the heart of quantum information science. While being initially only of philosophical interest, they now form the basis for device independent quantum cryptography [2,3] as well as for quantum advantage [4].…”