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.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.

arXiv:0811.0898Nov 6, 200831 citations

Classical simulation complexity of extended Clifford circuits

Richard Jozsa, M. Van den Nest

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.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.

Quantum Information and ComputationMay 1, 201433 citations

Embedding classical into quantum computation

Richard Jozsa

We describe a simple formalism for generating classes of quantum circuits that are classically efficiently simulatable and show that the efficient simulation of Clifford circuits (Gottesman-Knill theorem) and of matchgate circuits (Valiant's theorem) appear as two special cases. Viewing these simulatable classes as subsets of the space of all quantum computations, we may consider minimal extensions that suffice to regain full quantum computational power, which provides an approach to exploring the efficacy of quantum over classical computation.We describe a simple formalism for generating classes of quantum circuits that are classically efficiently simulatable and show that the efficient simulation of Clifford circuits (Gottesman-Knill theorem) and of matchgate circuits (Valiant's theorem) appear as two special cases. Viewing these simulatable classes as subsets of the space of all quantum computations, we may consider minimal extensions that suffice to regain full quantum computational power, which provides an approach to exploring the efficacy of quantum over classical computation.

arXiv:0812.4511Dec 24, 20086 citations

Classical simulation complexity of extended Clifford circuits

Richard Jozsa, Maarten Van den Nest

arXiv:1305.6190May 27, 20132 citations

Classical simulation complexity of extended clifford circuits

Richard Jozsa, M. Van den Nest

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 effciently 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 effciently 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. © Rinton Press.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 effciently 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 effciently 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. © Rinton Press.

Quantum Information and ComputationMay 1, 201435 citations

Clifford Simulation: Techniques and Applications

Alex Kerzner

Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. \n \nNext we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time $\\widetilde{O}(n^{\\omega/2})$ , where $\\omega < 2.373$ is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth- $d$ Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For $d= O(\\log n)$ , both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth- $d$ Clifford+ $T$ circuit with $t$ $T$ gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time $\\widetilde{O}(2^{0.7926t}n^{5/2}t^{6} d^3)$ and can compute output probabilities in time $\\widetilde{O}(2^{0.3963t}n^{3/2}t^{6} d^3)$ . Previous work [9,6], applied to the case $d=O(\\log n)$ , gives algorithms for sampling in time $O(2^{0.3963t} n^6 t^6)$ and computation of output probabilities in time $O(2^{0.3963t}n^3t^3)$ . Our sampling algorithm offers improved scaling in $n$ but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in $n$ with identical scaling in the exponential term.Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. \n \nNext we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time $\\widetilde{O}(n^{\\omega/2})$, where $\\omega < 2.373$ is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-$d$ Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For $d= O(\\log n)$, both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-$d$ Clifford+$T$ circuit with $t$ $T$ gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time $\\widetilde{O}(2^{0.7926t}n^{5/2}t^{6} d^3)$ and can compute output probabilities in time $\\widetilde{O}(2^{0.3963t}n^{3/2}t^{6} d^3)$. Previous work [9,6], applied to the case $d=O(\\log n)$, gives algorithms for sampling in time $O(2^{0.3963t} n^6 t^6)$ and computation of output probabilities in time $O(2^{0.3963t}n^3t^3)$. Our sampling algorithm offers improved scaling in $n$ but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in $n$ with identical scaling in the exponential term.

May 28, 20212 citations

\#\mathbf{P}

-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.

\\widetilde{O}(n^{\\omega/2})

, where

\\omega < 2.373

is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-

d

Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For

d= O(\\log n)

, both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-

d

Clifford+

T

circuit with

t

T

gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time

\\widetilde{O}(2^{0.7926t}n^{5/2}t^{6} d^3)

and can compute output probabilities in time

\\widetilde{O}(2^{0.3963t}n^{3/2}t^{6} d^3)

. Previous work [9,6], applied to the case

d=O(\\log n)

, gives algorithms for sampling in time

O(2^{0.3963t} n^6 t^6)

and computation of output probabilities in time

O(2^{0.3963t}n^3t^3)

. Our sampling algorithm offers improved scaling in

n

but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in

n

with identical scaling in the exponential term.