scholarly journals Further extensions of Clifford circuits and their classical simulation complexities

2017 ◽  
Vol 17 (3&4) ◽  
pp. 262-282
Author(s):  
Dax E. Koh
Keyword(s):  
Quantum Circuit ◽  
Complete Classification ◽  
Polynomial Hierarchy ◽  
New Combinations ◽  
Computational Power ◽  
Classical Simulation

Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations, which might just be slightly different, lead to circuits which are likely not. We extend the results of Jozsa and Van den Nest [Quant. Info. Comput. 14, 633 (2014)] by studying two further extensions of Clifford circuits. First, we consider how the classical simulation complexity changes when we allow for more general measurements. Second, we investigate different notions of what it means to ‘classically simulate’ a quantum circuit. These further extensions give us 24 new combinations of ingredients compared to Jozsa and Van den Nest, and we give a complete classification of their classical simulation complexities. Our results provide more examples where seemingly modest changes to the ingredients of Clifford circuits lead to “large” changes in the classical simulation complexities of the circuits, and also include new examples of extended Clifford circuits that exhibit “quantum supremacy”, in the sense that it is not possible to efficiently classically sample from the output distributions of such circuits, unless the polynomial hierarchy collapses.

scholarly journals Quantum advantage of unitary Clifford circuits with magic state inputs

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180427 ◽  
Author(s):  
Mithuna Yoganathan ◽  
Richard Jozsa ◽  
Sergii Strelchuk
Keyword(s):  
Polynomial Hierarchy ◽  
Efficient Computation ◽  
Computational Power ◽  
Average Case ◽  
Multiplicative Error ◽  
Additive Error ◽  
Universal Computation ◽  
Classical Simulation ◽  
Broad Variety ◽  
The Way

We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming polynomial hierarchy non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way, we give an extension of the Gottesman–Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabilizer and non-stabilizer inputs, the stabilizer part can be eliminated in favour of classical simulation, leaving a Clifford circuit on only the non-stabilizer part. Finally, we discuss implementational advantages of CM circuits.

scholarly journals Quantum advantage from energy measurements of many-body quantum systems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 465
Author(s):  
Leonardo Novo ◽  
Juani Bermejo-Vega ◽  
Raúl García-Patrón
Keyword(s):  
High Resolution ◽  
Measurement Errors ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
Many Body ◽  
Classical Simulation ◽  
Measurement Resolution ◽  
New Perspective ◽  
Sampling Problems

The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state – a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be exponentially fast-forwarded, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.

scholarly journals How many qubits are needed for quantum computational supremacy?

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 264 ◽  
Author(s):  
Alexander M. Dalzell ◽  
Aram W. Harrow ◽  
Dax Enshan Koh ◽  
Rolando L. La Placa
Keyword(s):  
Optical Networks ◽  
Quantum Computer ◽  
Circuit Simulation ◽  
Quantum Circuit ◽  
Deterministic Algorithm ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
Optical Elements ◽  
Average Case ◽  
Classical Simulation

Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P≠NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH(a) and per-int-NSETH(b) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F2 or the permanent of an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn time steps, where c∈{a,b}. A third conjecture poly3-ave-SBSETH(a′) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class SBP. We analyze evidence for these conjectures and argue that they are plausible when a=1/2, b=0.999 and a′=1/2.Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 104 to 107 gates.

Classical simulation of quantum computation, the gottesman-Knill theorem, and slightly beyond

2010 ◽  
Vol 10 (3&4) ◽  
pp. 258-271 ◽  
Author(s):  
M. Van den Nest
Keyword(s):  
Normal Form ◽  
Quantum Computation ◽  
Simple Proof ◽  
Quantum Circuit ◽  
Circuit Model ◽  
Computational Power ◽  
Classical Simulation ◽  
Starting Point

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 $\#\mathbf{P}$-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

Ricci inheritance collineations in Bianchi type II spacetime

2016 ◽  
Vol 31 (17) ◽  
pp. 1650102 ◽  
Author(s):  
Tahir Hussain ◽  
Sumaira Saleem Akhtar ◽  
Ashfaque H. Bokhari ◽  
Suhail Khan
Keyword(s):  
Lie Algebra ◽  
Ricci Tensor ◽  
Bianchi Type ◽  
Complete Classification ◽  
Type Ii

In this paper, we present a complete classification of Bianchi type II spacetime according to Ricci inheritance collineations (RICs). The RICs are classified considering cases when the Ricci tensor is both degenerate as well as non-degenerate. In case of non-degenerate Ricci tensor, it is found that Bianchi type II spacetime admits 4-, 5-, 6- or 7-dimensional Lie algebra of RICs. In the case when the Ricci tensor is degenerate, majority cases give rise to infinitely many RICs, while remaining cases admit finite RICs given by 4, 5 or 6.

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