scholarly journals Classical simulation complexity of extended Clifford circuits

2014 ◽  
Vol 14 (7&8) ◽  
pp. 633-648
Author(s):  
Richard Jozsa ◽  
Marrten Van den Nest
Keyword(s):  
Quantum Computing ◽  
Physical Significance ◽  
Quantum Circuits ◽  
Quantum Processes ◽  
Computing Power ◽  
Quantum Operations ◽  
Classical Simulation ◽  
Universal Quantum ◽  
General Product ◽  
Computational Basis

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.

scholarly journals Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 106 ◽  
Author(s):  
Tomoyuki Morimae ◽  
Yuki Takeuchi ◽  
Harumichi Nishimura
Keyword(s):  
Quantum Computing ◽  
Polynomial Time ◽  
Probability Distributions ◽  
Quantum Circuits ◽  
Universal Models ◽  
Reversible Circuit ◽  
Universal Quantum ◽  
Quantum Polynomial ◽  
Output Probability ◽  
Probabilistic Polynomial Time

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.

scholarly journals Matchgates and classical simulation of quantum circuits

2008 ◽  
Vol 464 (2100) ◽  
pp. 3089-3106 ◽  
Author(s):  
Richard Jozsa ◽  
Akimasa Miyake
Keyword(s):  
Clifford Algebra ◽  
Point Of View ◽  
Quantum Circuits ◽  
Nearest Neighbour ◽  
Computational Power ◽  
Classical Simulation ◽  
Wigner Representation ◽  
Universal Quantum ◽  
The One ◽  
Qubit Gate

Let G ( A ,  B ) denote the two-qubit gate that acts as the one-qubit SU (2) gates A and B in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan–Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.

scholarly journals QUANTUM CIRCUITS FOR SINGLE-QUBIT MEASUREMENTS CORRESPONDING TO PLATONIC SOLIDS

2004 ◽  
Vol 02 (03) ◽  
pp. 353-377 ◽  
Author(s):  
THOMAS DECKER ◽  
DOMINIK JANZING ◽  
THOMAS BETH
Keyword(s):  
Fourier Transform ◽  
Quantum Circuits ◽  
Platonic Solid ◽  
Bloch Sphere ◽  
Platonic Solids ◽  
Quantum Operations ◽  
Single Qubit ◽  
Orthogonal Measurement ◽  
Computational Basis ◽  
Positive Operator Valued Measure

Each platonic solid defines a single-qubit positive operator-valued measure (POVM) by interpreting its vertices as points on the Bloch sphere. We construct simple circuits for implementing these kinds of measurements and other simple types of symmetric POVMs on one qubit. Each implementation consists of a discrete Fourier transform and some elementary quantum operations followed by an orthogonal measurement in the computational basis.

Both Toffoli and Controlled-NOT need little help to universal quantum computing

2003 ◽  
Vol 3 (1) ◽  
pp. 84-92
Author(s):  
Y-Y Shi
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Circuit ◽  
Single Qubit ◽  
Universal Quantum ◽  
Qubit Gate ◽  
Computational Basis ◽  
Universal Gates

What additional gates are needed for a set of classical universal gates to do universal quantum computation? We prove that any single-qubit real gate suffices, except those that preserve the computational basis. The Gottesman-Knill Theorem implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its like) are universal for quantum computing. Previously only a generic gate, namely a rotation by an angle incommensurate with \pi, is known to be sufficient in both problems, if only one single-qubit gate is added.

scholarly journals Quantum circuits and Spin(3n) groups

2015 ◽  
Vol 15 (3&4) ◽  
pp. 235-259
Author(s):  
Alexander Yu. Vlasov
Keyword(s):  
Tensor Product ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Spin Groups ◽  
Classical Simulation ◽  
The Matrix ◽  
Universal Quantum ◽  
Usual Quantum

All quantum gates with one and two qubits may be described by elements of Spin groups due to isomorphisms Spin(3)\isomSU(2) and Spin(6)\isomSU(4). However, the group of n-qubit gates SU(2^n) for n>2 has bigger dimension than Spin(3n). A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation SU(2^n). Analogously, the `$Spin(3n)$ circuits' are introduced in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms. The matrix tensor product implementation of the Spin(3n) group together with relevant models by usual quantum circuits with 2n qubits are investigated in such a framework. A certain resemblance with well-known sets of non-universal quantum gates (e.g., matchgates, noninteracting-fermion quantum circuits) related with Spin(2n) may be found in presented approach. Finally, a possibility of the classical simulation of such circuits in polynomial time is discussed.

Universal quantum circuit for n-qubit quantum gate: a programmable quantum gate

2007 ◽  
Vol 7 (3) ◽  
pp. 228-242
Author(s):  
P.B.M. Sousa ◽  
R.V. Ramos
Keyword(s):  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Gate ◽  
Quantum Circuits ◽  
Unitary Matrices ◽  
Quantum Operations ◽  
Universal Quantum ◽  
Np Problems ◽  
Universal Quantum Circuit ◽  
Specific Quantum

Quantum computation has attracted much attention, among other things, due to its potentialities to solve classical NP problems in polynomial time. For this reason, there has been a growing interest to build a quantum computer. One of the basic steps is to implement the quantum circuit able to realize a given unitary operation. This task has been solved using decomposition of unitary matrices in simpler ones till reach quantum circuits having only single-qubits and CNOTs gates. Usually the goal is to find the minimal quantum circuit able to solve a given problem. In this paper we go in a different direction. We propose a general quantum circuit able to implement any specific quantum circuit by just setting correctly the parameters. In other words, we propose a programmable quantum circuit. This opens the possibility to construct a real quantum computer where several different quantum operations can be realized in the same hardware. The configuration is proposed and its optical implementation is discussed.

scholarly journals Efficient Classical Simulation and Benchmarking of Quantum Processes in the Weyl Basis

2021 ◽  
Vol 126 (21) ◽  
Author(s):  
Daniel Stilck França ◽  
Sergii Strelchuk ◽  
Michał Studziński
Keyword(s):  
Quantum Processes ◽  
Classical Simulation ◽  
Weyl Basis

scholarly journals Efficient Quantum Circuits for One-Way Quantum Computing

2009 ◽  
Vol 102 (10) ◽  
Author(s):  
Tetsufumi Tanamoto ◽  
Yu-xi Liu ◽  
Xuedong Hu ◽  
Franco Nori
Keyword(s):  
Quantum Computing ◽  
Quantum Circuits

Optimal two-particle entanglement by universal quantum processes

2001 ◽  
Vol 1 (3) ◽  
pp. 33-51
Author(s):  
G Alber ◽  
A Delgado ◽  
I Jex
Keyword(s):  
Hilbert Spaces ◽  
Parameter Family ◽  
Entanglement Measure ◽  
Input State ◽  
Quantum Processes ◽  
Single Exception ◽  
Universal Quantum

Within the class of all possible universal (covariant) two-particle quantum processes in arbitrary dimensional Hilbert spaces those universal quantum processes are determined whose output states optimize the recently proposed entanglement measure of Vidal and Werner. It is demonstrated that these optimal entanglement processes belong to a one-parameter family of universal entanglement processes whose output states do not contain any separable components. It is shown that these optimal universal entanglement processes generate antisymmetric output states and, with the single exception of qubit systems, they preserve information about the initial input state.

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