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

Complexity Classification of Conjugated Clifford Circuits

Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets

Principle of majorization: Application to random quantum circuits

We test the principle of majorization [J. I. Latorre and M. A. Martín-Delgado, Phys. Rev. A 66, 022305 (2002)] in random circuits. Three classes of circuits were considered: (i) universal, (ii) classically simulatable, and (iii) neither universal nor classically simulatable. The studied families are: {CNOT, H, T}, {CNOT, H, NOT}, {CNOT, H, S} (Clifford), matchgates, and IQP (instantaneous quantum polynomial-time). We verified that all the families of circuits satisfy on average the principle of decreasing majorization. In most cases the asymptotic state (number of gates → ∞) behaves like a random vector. However, clear differences appear in the fluctuations of the Lorenz curves associated to asymptotic states. The fluctuations of the Lorenz curves discriminate between universal and non-universal classes of random quantum circuits, and they also detect the complexity of some non-universal but not classically efficiently simulatable quantum random circuits. We conclude that majorization can be used as a indicator of complexity of quantum dynamics, as an alternative to, e.g., entanglement spectrum and out-of-time-order correlators (OTOCs).Fil: Vallejos, Raúl O.. Centro Brasileiro de Pesquisas Físicas; BrasilFil: De Melo, Fernando. Centro Brasileiro de Pesquisas Físicas; BrasilFil: Carlo, Gabriel Gustavo. Comisión Nacional de Energía Atómica. Gerencia de Área Investigaciones y Aplicaciones No Nucleares. Gerencia Física (CAC). Departamento de Física de la Materia Condensada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin