49 research outputs found
Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond
We study classical simulation of quantum computation, taking the
Gottesman-Knill theorem as a starting point. We show how each Clifford circuit
can be reduced to an equivalent, manifestly simulatable circuit (normal form).
This provides a simple proof of the Gottesman-Knill theorem without resorting
to stabilizer techniques. The normal form highlights why Clifford circuits have
such limited computational power in spite of their high entangling power. At
the same time, the normal form shows how the classical simulation of Clifford
circuits fits into the standard way of embedding classical computation into the
quantum circuit model. This leads to simple extensions of Clifford circuits
which are classically simulatable. These circuits can be efficiently simulated
by classical sampling ('weak simulation') even though the problem of exactly
computing the outcomes of measurements for these circuits ('strong simulation')
is proved to be #P-complete--thus showing that there is a separation between
weak and strong classical simulation of quantum computation.Comment: 14 pages, shortened version, one additional result. To appear in
Quant. Inf. Com
Simulating quantum circuit expectation values by Clifford perturbation theory
The classical simulation of quantum circuits is of central importance for
benchmarking near-term quantum devices. The fact that gates belonging to the
Clifford group can be simulated efficiently on classical computers has
motivated a range of methods that scale exponentially only in the number of
non-Clifford gates. Here, we consider the expectation value problem for
circuits composed of Clifford gates and non-Clifford Pauli rotations, and
introduce a heuristic perturbative approach based on the truncation of the
exponentially growing sum of Pauli terms in the Heisenberg picture. Numerical
results are shown on a Quantum Approximate Optimization Algorithm (QAOA)
benchmark for the E3LIN2 problem and we also demonstrate how this method can be
used to quantify coherent and incoherent errors of local observables in
Clifford circuits. Our results indicate that this systematically improvable
perturbative method offers a viable alternative to exact methods for
approximating expectation values of large near-Clifford circuits
Classical simulation complexity of extended Clifford circuits
Clifford gates are a winsome class of quantum operations combining
mathematical elegance with physical significance. The Gottesman-Knill theorem
asserts that Clifford computations can be classically efficiently simulated but
this is true only in a suitably restricted setting. Here we consider Clifford
computations with a variety of additional ingredients: (a) strong vs. weak
simulation, (b) inputs being computational basis states vs. general product
states, (c) adaptive vs. non-adaptive choices of gates for circuits involving
intermediate measurements, (d) single line outputs vs. multi-line outputs. We
consider the classical simulation complexity of all combinations of these
ingredients and show that many are not classically efficiently simulatable
(subject to common complexity assumptions such as P not equal to NP). Our
results reveal a surprising proximity of classical to quantum computing power
viz. a class of classically simulatable quantum circuits which yields universal
quantum computation if extended by a purely classical additional ingredient
that does not extend the class of quantum processes occurring.Comment: 17 pages, 1 figur
Just Like the Real Thing: Fast Weak Simulation of Quantum Computation
Quantum computers promise significant speedups in solving problems
intractable for conventional computers but, despite recent progress, remain
limited in scaling and availability. Therefore, quantum software and hardware
development heavily rely on simulation that runs on conventional computers.
Most such approaches perform strong simulation in that they explicitly compute
amplitudes of quantum states. However, such information is not directly
observable from a physical quantum computer because quantum measurements
produce random samples from probability distributions defined by those
amplitudes. In this work, we focus on weak simulation that aims to produce
outputs which are statistically indistinguishable from those of error-free
quantum computers. We develop algorithms for weak simulation based on quantum
state representation in terms of decision diagrams. We compare them to using
state-vector arrays and binary search on prefix sums to perform sampling.
Empirical validation shows, for the first time, that this enables mimicking of
physical quantum computers of significant scale.Comment: 6 pages, 4 figure
Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable
We study the classical simulatability of commuting quantum circuits with n
input qubits and O(log n) output qubits, where a quantum circuit is classically
simulatable if its output probability distribution can be sampled up to an
exponentially small additive error in classical polynomial time. First, we show
that there exists a commuting quantum circuit that is not classically
simulatable unless the polynomial hierarchy collapses to the third level. This
is the first formal evidence that a commuting quantum circuit is not
classically simulatable even when the number of output qubits is exponentially
small. Then, we consider a generalized version of the circuit and clarify the
condition under which it is classically simulatable. Lastly, we apply the
argument for the above evidence to Clifford circuits in a similar setting and
provide evidence that such a circuit augmented by a depth-1 non-Clifford layer
is not classically simulatable. These results reveal subtle differences between
quantum and classical computation.Comment: 19 pages, 6 figures; v2: Theorems 1 and 3 improved, proofs modifie