Checking whether two quantum circuits are approximately equivalent is a common task in quantum computing. We consider a closely related identity check problem: given a quantum circuit $U$, one has to estimate the diamond-norm distance between $U$ and the identity channel. We present a classical algorithm approximating the distance to the identity within a factor $α=D+1$ for shallow geometrically local $D$-dimensional circuits provided that the circuit is sufficiently close to the identity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit depth and spatial dimension. We also show that the operator-norm distance to the identity $\|U-I\|$ can be efficiently approximated within a factor $α=5$ for shallow 1D circuits and, under a certain technical condition, within a factor $α=2D+3$ for shallow $D$-dimensional circuits. A numerical implementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100 qubits.

An n-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of n. We describe a classical algorithm with quasipolynomial runtime nO(logn) that approximately samples from the output distribution of a peaked constant-depth circuit. We give even faster algorithms for circuits composed of nearest-neighbor gates on a D-dimensional grid of qubits, with polynomial runtime nO(1) if D=2 and almost-polynomial runtime nO(loglogn) for D>2. Our sampling algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error, improving previously known methods. As a simple application, we obtain a quasipolynomial algorithm to estimate the magnitude of the expected value of any Pauli observable in the output state of a shallow circuit (which may or may not be peaked). This is a dramatic improvement over the prior state-of-the-art algorithm which had an exponential scaling in √n.

The Hidden Weighted Bit function plays an important role in the study of classical models of computation. A common belief is that this function is exponentially hard to implement using reversible ancilla-free circuits, even though introducing a small number of ancillae allows a very efficient implementation. In this paper, we refute the exponential hardness conjecture by developing a polynomial-size reversible ancilla-free circuit computing the Hidden Weighted Bit function. Our circuit has size O(n^6.42), where n is the number of input bits. We also show that the Hidden Weighted Bit function can be computed by a quantum ancilla-free circuit of size O(n^2). The technical tools employed come from a combination of Theoretical Computer Science (Barringtons theorem) and Physics (simulation of fermionic Hamiltonians) techniques.

The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form F1HSF2, where H is a layer of Hadamard gates, S is a permutation of qubits, and Fiare parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed n-qubit Clifford operator in runtime O(n2). The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth 9n implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

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. 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 constant-depth 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.

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most O(√n) qubits, where n is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight O(nα) for any α > 0.

Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L1-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L1-distance ∥p-q∥1can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).

We derive new tradeoffs for reliable quantum information storage in a 2D local architecture based on subsystem quantum codes. Our results apply to stabilizer subsystem codes, that is, stabilizer codes in which part of the logical qubits does not encode any information. A stabilizer subsystem code can be specified by its gauge group - a subgroup of the Pauli group that includes the stabilizers and the logical operators on the unused logical qubits. We assume that the physical qubits are arranged on a two-dimensional grid and the gauge group has spatially local generators such that each generator acts only on a few qubits located close to each other. Our main result is an upper bound kd = O(n), where k is the number of encoded qubits, d is the minimal distance, and n is the number of physical qubits. In the special case when both gauge group and the stabilizer group have spatially local generators, we derive a stronger bound kd2= O(n) which is tight up to a constant factor.

Suppose one has access to oracles generating samples from two unknown probability distributions $p$ and $q$ on some $N$-element set. How many samples does one need to test whether the two distributions are close or far from each other in the $L_1$-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the $L_1$-distance $\|p-q\|_1$ can be estimated with a constant precision using only $O(N^{1/2})$ queries in the quantum settings, whereas classical computers need $\Omega(N^{1-o(1)})$ queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity $O(N^{1/3})$. The classical query complexity of these problems is known to be $\Omega(N^{1/2})$. A quantum algorithm for testing Uniformity has been recently independently discovered by Chakraborty et al. \cite{CFMW09}.

We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as $H=\sum_{a}H_{a}$ where (1) every term $H_{a}$ acts nontrivially on a constant number of qubits, (2) every term $H_{a}$ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of H is a ground-state of every term $H_{a}$. Second, we generalize the Cook–Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin–Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the k-SAT problem which we call “stoquastic k-SAT” such that stoquastic k-SAT is contained in MA for any constant k, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.