A large class of quantum algorithms can be represented by placing a reversible classical circuit, or diagonal gates, between a pair of quantum Fourier transformations. Notable examples include the circuit from Shor's factoring algorithm and IQP circuits. Here we study quantum circuits with a similar structure, where the QFT has been replaced by a sequentially coupled quantum Schur transform. We extend the Kushilevitz-Mansour algorithm for sampling from Fourier-sparse output distributions to these circuits. It is shown that samples from a classical distribution close to the output of a sparse Schur circuit can be efficiently generated by this classical method. In particular, it implies that circuits as found in a computationally interesting regime of Permutational Quantum Computing can be sampled from efficiently classically, assuming they give rise to sparse output distributions. Numerical simulations indicate that a significant fraction of Permutational circuits on a small number of qubits have this property, allowing for efficient sampling of their output.
We show that several quantum circuit families can be simulated efficiently classically if it is promised that their output distribution is approximately sparse i.e. the distribution is close to one where only a polynomially small, a priori unknown subset of the measurement probabilities are nonzero. Classical simulations are thereby obtained for quantum circuits which---without the additional sparsity promise---are considered hard to simulate. Our results apply in particular to a family of Fourier sampling circuits (which have structural similarities to Shor's factoring algorithm) but also to several other circuit families, such as IQP circuits. Our results provide examples of quantum circuits that cannot achieve exponential speed-ups due to the presence of too much destructive interference i.e. too many cancelations of amplitudes. The crux of our classical simulation is an efficient algorithm for approximating the significant Fourier coefficients of a class of states called computationally tractable states. The latter result may have applications beyond the scope of this work. In the proof we employ and extend sparse approximation techniques, in particular the Kushilevitz-Mansour algorithm, in combination with probabilistic simulation methods for quantum circuits.
Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the SU(2) quantum Schur transform---a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which they could lead to algorithms with exponential computational advantages. Our paper is primarily motivated by a conjecture that underpinned the hardness of permutational quantum computing, a restricted quantum computational model that has the above circuit structure in one of its computationally interesting regimes. The conjecture stated that approximating transition amplitudes of permutational quantum computing model to inverse-polynomial precision on a classical computer is computationally hard in this case. We disprove the extended version of this conjecture---even in the case when the hardness of approximation originated from a difficulty of finding the large elements in the output probability distributions. Finally, we present some evidence that output of the above permutational quantum computing circuits could be efficiently approximately sampled from on a classical computer.
We study the classical simulability of a polynomial-size quantum circuit
Cn on
n qubits followed by sparse classical post-processing (SCP) on
m bits, where
m≤n≤poly(m). The SCP is described by a non-zero Boolean function
fm that is classically computable in polynomial time and is sparse, i.e., has a peaked Fourier spectrum. First, we provide a necessary and sufficient condition on
Cn such that, for any SCP
fm,
Cn followed by
fm is classically simulable. This characterization extends the result of Van den Nest and implies that various quantum circuits followed by SCP are classically simulable. Examples include IQP circuits, Clifford Magic circuits, and the quantum part of Simon's algorithm, even though these circuits alone are hard to simulate classically. Then, we consider the case where
Cn has constant depth
d. While it is unlikely that, for any SCP
fm,
Cn followed by
fm is classically simulable, we show that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on
n+1 qubits. Each such circuit consists of at most deg(
fm) commuting gates and each commuting gate acts on at most
2d+1 qubits, where deg(
fm) is the Fourier degree of
fm. This provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.
Permutational Quantum Computing (Jordan, 2009) is a natural quantum computational model that was conjectured to capture non-classical aspects of quantum computation. Contrary to the previous conjecture, we find a classical algorithm that efficiently approximates output distributions of the model up to polynomially small additive precision. We extend this algorithm to show that a large class of important quantum circuits - the Quantum Schur Sampling circuits - can be also efficiently simulated classically. The algorithm can be used to efficiently estimate non-trivial elements of irreducible representation matrices of the symmetric group on
n elements and might be also used to approximate transition amplitudes of the Ponzano-Regge model.
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and
exp[iθX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.