By the Gottesman-Knill Theorem, the outcome probabilities of Clifford circuits can be computed efficiently. We present an alternative proof of this result for quopit Clifford circuits (i.e., Clifford circuits on collections of p-level systems, where p is an odd prime) using Feynman’s sum-over-paths technique, which allows the amplitudes of arbitrary quantum circuits to be expressed in terms of a weighted sum over computational paths. For a general quantum circuit, the sum over paths contains an exponential number of terms, and no efficient classical algorithm is known that can compute the sum. For quopit Clifford circuits, however, we show that the sum over paths takes a special form: it can be expressed as a product of Weil sums with quadratic polynomials, which can be computed efficiently. This provides a method for computing the outcome probabilities and amplitudes of such circuits efficiently, and is an application of the circuit-polynomial correspondence which relates quantum circuits to low-degree polynomials.
Strongly simulating a quantum circuit, that is, computing an output amplitude, can be done by summing the circuit's Feynman paths: a weighted count over assignments to Boolean path variables. The circuit's gates induce correlations among these variables, forming a graph whose structure controls several exact simulation routes. This sum-of-powers (SOP) viewpoint underlies recent simulators built on binary decision diagrams and weighted model counting.
For a quadratic SOP with
n variables, even modulus
r, and a rank-decomposition of its variable graph of width
k, our dynamic program (DP) computes an amplitude using only
O(4kpoly(n)) arithmetic operations. For Clifford
+T circuits, the amplitude is given by an SOP with modulus
8.
Rank-width never exceeds linear rank-width, which governs some decision-diagram approaches, and is at most one greater than the Markov--Shi contraction complexity of the circuit tensor network. Moreover, there are non-Clifford families of bounded rank-width where both competing parameters diverge. We also present a stabilizer-rank optimization, exploiting that the DP tables are stabilizer-type Gauss sums. Each subtree runs at the width price of its cut-ranks or at a magic price that discharges the non-Clifford phases below it. The resulting best total cost never exceeds
O(4kpoly(n)), yet is polynomial on mixed families where the pure rank-width and pure
T-count guarantees are both exponential. Clifford amplitudes take polynomial time on any graph, the exact-amplitude consequence of Gottesman--Knill.
A prototype evaluation on standard circuit benchmarks finds treewidth bucket elimination the strongest baseline, with the new rank-width DP complementary: it wins on structured families where treewidth blows up.
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
widetildeO(nomega/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(logn), 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
widetildeO(20.7926tn5/2t6d3) and can compute output probabilities in time
widetildeO(20.3963tn3/2t6d3). Previous work [9,6], applied to the case
d=O(logn), gives algorithms for sampling in time
O(20.3963tn6t6) and computation of output probabilities in time
O(20.3963tn3t3). 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.
We introduce and analyze circuits that are the quantum mechanical generalization of classical algebraic circuits. Using the algebraic operations of addition and multiplication, as well as the quantum Fourier transform, such circuits are well-defined for rings Z=mZ and finite fields Fq. The acceptance probabilities of such algebraic quantum circuits can be expressed as exponential sums Âx exp(2pi f(x)=m) where the multivariate polynomial f is determined by the circuit, while it is independent of the ring or field over which we interpret the circuit. Dawson et al. [Quantum Information & Computation, 5(2), pp. 102‐112 (2004)] introduced this “sum over paths” description as a discrete version of the path integral approach of standard quantum mechanics. From this perspective, the polynomial f should be interpreted as the “action” of a specific (classical) computational path between the input and output of the circuit. In this article we prove several properties of algebraic quantum circuits. Using the theory of exponential sums, we show that in the limit of large m or q, the acceptance probabilities of a circuit converge to zero or to one. Circuits that do not involve the multiplication operation are the algebraic generalization of Clifford circuits and we show how their acceptance probabilities can be calculated exactly in a classical efficient manner. For algebraic circuits that are defined over rings Z=p r Z we derive a “least action principle” that shows how the behaviour of such circuits is determined by those computational paths whose action polynomials are extremal.
Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In \cite{VDNest_12_QFTs} it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.
Path sums are a convenient symbolic formalism for quantum operations with applications to the simulation, optimization, and verification of quantum protocols.Unlike quantum circuits, path sums are not limited to unitary operations, but can express arbitrary linear ones.Two problems, therefore, naturally arise in the study of path sums: the unitarity problem and the extraction problem.The former is the problem of deciding whether a given path sum represents a unitary operator.The latter is the problem of constructing a quantum circuit, given a path sum promised to represent a unitary operator.In this paper, we show that the unitarity problem is co-NP-hard in general, but that it is in P when restricted to Clifford path sums.We then provide an algorithm to synthesize a Clifford circuit from a unitary Clifford path sum.The circuits produced by our extraction algorithm are of the form C 1 HC 2 , where C 1 and C 2 are Hadamard-free circuits and H is a layer of Hadamard gates.We also provide a heuristic generalization of our extraction algorithm to arbitrary path sums.While this algorithm is not guaranteed to succeed, it often succeeds and typically produces natural looking circuits.Alongside applications to the optimization and decompilation of quantum circuits, we demonstrate the capability of our algorithm by synthesizing the standard quantum Fourier transform directly from a path sum.
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.