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 Efficient universal quantum circuits

2010 ◽  
Vol 10 (1&2) ◽  
pp. 16-27
Author(s):  
D. Bera ◽  
S. Fenner ◽  
F. Green ◽  
S. Homer
Keyword(s):  
Blow Up ◽  
General Purpose ◽  
Quantum Circuits ◽  
Computational Power ◽  
Universal Quantum

Universal circuits can be viewed as general-purpose simulators for central classes of circuits and can be used to capture the computational power of the circuit class being simulated. We define and construct quantum universal circuits which are efficient and has very little overhead in simulation. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here which is nearly optimal.

scholarly journals Efficient classical simulation of noisy random quantum circuits in one dimension

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 318 ◽  
Author(s):  
Kyungjoo Noh ◽  
Liang Jiang ◽  
Bill Fefferman
Keyword(s):  
Error Rate ◽  
Entanglement Entropy ◽  
System Size ◽  
Quantum Circuits ◽  
One Dimension ◽  
Matrix Product ◽  
Computational Power ◽  
Characteristic System ◽  
Classical Simulation ◽  
The Cost

Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.

scholarly journals From estimation of quantum probabilities to simulation of quantum circuits

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 223 ◽  
Author(s):  
Hakop Pashayan ◽  
Stephen D. Bartlett ◽  
David Gross
Keyword(s):  
Quantum System ◽  
Additional Assumption ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Computational Power ◽  
Quantum Probabilities ◽  
Output Distribution ◽  
Classical Simulation ◽  
Classical Computer ◽  
Promising Avenue

Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of ``classical simulation'' and in particular on the required accuracy. We argue that a notion of classical simulation, which we call EPSILON-simulation (or ϵ-simulation for short), captures the essence of possessing ``equivalent computational power'' as the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an ϵ-simulator from one possessing the simulated quantum system. We relate ϵ-simulation to various alternative notions of simulation predominantly focusing on a simulator we call a poly-box. A poly-box outputs 1/poly precision additive estimates of Born probabilities and marginals. This notion of simulation has gained prominence through a number of recent simulability results. Accepting some plausible computational theoretic assumptions, we show that ϵ-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to ϵ-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity).

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.

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.

Understanding functional illiteracy from a policy, adult education, and cognition point of view: Towards a joint referent framework

2019 ◽  
Vol 30 (2) ◽  
pp. 109-122
Author(s):  
Aleksandar Bulajić ◽  
Miomir Despotović ◽  
Thomas Lachmann
Keyword(s):  
Adult Education ◽  
Operational Definition ◽  
Literacy Skills ◽  
Point Of View ◽  
Industrialized Countries ◽  
Functional Illiteracy ◽  
Scale Levels ◽  
Components Of Reading ◽  
The One ◽  
Definition Of

Abstract. The article discusses the emergence of a functional literacy construct and the rediscovery of illiteracy in industrialized countries during the second half of the 20th century. It offers a short explanation of how the construct evolved over time. In addition, it explores how functional (il)literacy is conceived differently by research discourses of cognitive and neural studies, on the one hand, and by prescriptive and normative international policy documents and adult education, on the other hand. Furthermore, it analyses how literacy skills surveys such as the Level One Study (leo.) or the PIAAC may help to bridge the gap between cognitive and more practical and educational approaches to literacy, the goal being to place the functional illiteracy (FI) construct within its existing scale levels. It also sheds more light on the way in which FI can be perceived in terms of different cognitive processes and underlying components of reading. By building on the previous work of other authors and previous definitions, the article brings together different views of FI and offers a perspective for a needed operational definition of the concept, which would be an appropriate reference point for future educational, political, and scientific utilization.

Recommendations of The Ombudsman’s Dilemma: Between Law Enforcement Officials and Protection of Credibility

2018 ◽  
Vol 2 ◽  
pp. 1-12
Author(s):  
Dyah Adriantini Sintha Dewi
Keyword(s):  
Law Enforcement ◽  
Public Service ◽  
Public Services ◽  
Point Of View ◽  
Law Enforcement Officials ◽  
Muslim Majority ◽  
The Republic ◽  
The Government ◽  
The One ◽  
Oversight Body

The Ombudsman as an external oversight body for official performance, in Fikih Siyasah (constitutionality in Islam) is included in the supervision stipulated in legislation (al-musahabah al-qomariyah). Supervision is done so that public service delivery to the community is in accordance with the rights of the community. This is done because in carrying out its duties, officials are very likely to conduct mal administration, which is bad public services that cause harm to the community. The Ombudsman is an institution authorized to resolve the mal administration issue, in which one of its products is by issuing a recommendation. Although Law No. 37 of 2018 on the Ombudsman of the Republic of Indonesia states that the recommendation is mandatory, theombudsman's recommendations have not been implemented. This is due to differences in point of view, ie on the one hand in the context of law enforcement, but on the other hand the implementation of the recommendation is considered as a means of opening the disgrace of officials. Recommendations are the last alternative of Ombudsman's efforts to resolve the mal administration case, given that a win-win solution is the goal, then mediation becomes the main effort. This is in accordance with the condition of the Muslim majority of Indonesian nation and prioritizes deliberation in resolving dispute. Therefore, it is necessary to educate the community and officials related to the implementation of the Ombudsman's recommendations in order to provide good public services for the community, which is the obligation of the government.

Defending the Difference

2017 ◽  
Keyword(s):  
Social Sciences ◽  
Natural Sciences ◽  
Point Of View ◽  
True Nature ◽  
Natural Entities ◽  
The Social ◽  
The World ◽  
Methodological Difference ◽  
The Difference ◽  
The One

Dreyfus argues that there is a basic methodological difference between the natural sciences and the social sciences, a difference that derives from the different goals and practices of each. He goes on to argue that being a realist about natural entities is compatible with pluralism or, as he calls it, “plural realism.” If intelligibility is always grounded in our practices, Dreyfus points out, then there is no point of view from which one can ask about or provide an answer to the one true nature of ultimate reality. But that is consistent with believing that the natural sciences can still reveal the way the world is independent of our theories and practices.

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

Sign in / Sign up

Export Citation Format

Share Document