On the computational power of constant-depth quantum circuits with gates for addition

2004 ◽  
Author(s):  
Y. Takahashi ◽  
Y. Kawano ◽  
M. Kitagawa
Keyword(s):  
Quantum Circuits ◽  
Computational Power ◽  
Constant Depth

Commuting quantum circuits: efficiently classical simulations versus hardness results

2013 ◽  
Vol 13 (1&2) ◽  
pp. 54-72
Author(s):  
Xiaotong Ni ◽  
Maarten van den Nest
Keyword(s):  
Sampling Methods ◽  
Quantum Effects ◽  
High Accuracy ◽  
Quantum Circuits ◽  
Computational Power ◽  
Restricted Class ◽  
Constant Depth ◽  
Single Qubit ◽  
Single Qudit ◽  
Local Circuits

The study of quantum circuits composed of commuting gates is particularly useful to understand the delicate boundary between quantum and classical computation. Indeed, while being a restricted class, commuting circuits exhibit genuine quantum effects such as entanglement. In this paper we show that the computational power of commuting circuits exhibits a surprisingly rich structure. First we show that every 2-local commuting circuit acting on $d$-level systems and followed by single-qudit measurements can be efficiently simulated classically with high accuracy. In contrast, we prove that such strong simulations are hard for 3-local circuits. Using sampling methods we further show that all commuting circuits composed of exponentiated Pauli operators $e^{i\theta P}$ can be simulated efficiently classically when followed by single-qubit measurements. Finally, we show that commuting circuits can efficiently simulate certain non-commutative processes, related in particular to constant-depth quantum circuits. This gives evidence that the power of commuting circuits goes beyond classical computation.

scholarly journals Comparing quantum hybrid reinforcement learning to classical methods

2021 ◽  
Author(s):  
Maximilian Moll ◽  
Leonhard Kunczik
Keyword(s):  
Reinforcement Learning ◽  
Quantum Circuits ◽  
Memory Usage ◽  
Computational Power ◽  
Classical Domain ◽  
Q Learning ◽  
Optimal Behavior ◽  
Computational Speed ◽  
Complex Decision ◽  
Hybrid Reinforcement

AbstractIn recent history, reinforcement learning (RL) proved its capability by solving complex decision problems by mastering several games. Increased computational power and the advances in approximation with neural networks (NN) paved the path to RL’s successful applications. Even though RL can tackle more complex problems nowadays, it still relies on computational power and runtime. Quantum computing promises to solve these issues by its capability to encode information and the potential quadratic speedup in runtime. We compare tabular Q-learning and Q-learning using either a quantum or a classical approximation architecture on the frozen lake problem. Furthermore, the three algorithms are analyzed in terms of iterations until convergence to the optimal behavior, memory usage, and runtime. Within the paper, NNs are utilized for approximation in the classical domain, while in the quantum domain variational quantum circuits, as a quantum hybrid approximation method, have been used. Our simulations show that a quantum approximator is beneficial in terms of memory usage and provides a better sample complexity than NNs; however, it still lacks the computational speed to be competitive.

CONSTANT-DEPTH PERIODIC CIRCUITS

1991 ◽  
Vol 01 (01) ◽  
pp. 49-87 ◽  
Author(s):  
HOWARD STRAUBING
Keyword(s):  
Solvable Groups ◽  
Computational Power ◽  
Constant Depth ◽  
Branching Programs ◽  
Dihedral Groups ◽  
Circuit Element ◽  
Independent Evidence ◽  
Polynomial Size ◽  
And Function ◽  
Fixed Period

This paper is devoted to the languages accepted by constant-depth, polynomial-size families of circuits in which every circuit element computes the sum of its input bits modulo a fixed period q. It has been conjectured that such a circuit family cannot compute the AND function of n inputs. Here it is shown that such circuit families are equivalent in power to polynomial-length programs over finite solvable groups; in particular, the conjecture implies that Barrington's result on the computational power of branching programs over nonsolvable groups cannot be extended to solvable groups. It is also shown that polynomial-length programs over dihedral groups cannot compute the AND function. Furthermore, it is shown that the conjecture is equivalent to a characterization, in terms of finite semigroups and formal logic, of the regular languages accepted by such circuit families. There is, moreover, considerable independent evidence for this characterization. This last result is established using new theorems, of independent interest, concerning the algebraic structure of finite categories.

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 Adaptive Quantum Computation, Constant Depth Quantum Circuits and Arthur-Merlin Games

2004 ◽  
Vol 4 (2) ◽  
pp. 134-145 ◽  
Author(s):  
B.M. Terhal ◽  
D.P. DiVincenzo
Keyword(s):  
Quantum Computation ◽  
High Accuracy ◽  
Quantum Circuits ◽  
Constant Depth ◽  
Present Evidence ◽  
Quantum Computations

We present evidence that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy. We prove that if one can simulate these circuits classically efficiently then ${\rm BQP} \subseteq {\rm AM}$.

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 Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

2016 ◽  
Vol 25 (4) ◽  
pp. 849-881 ◽  
Author(s):  
Yasuhiro Takahashi ◽  
Seiichiro Tani
Keyword(s):  
Quantum Circuits ◽  
Constant Depth ◽  
Exact Quantum

Quantum lower bounds for fanout

2006 ◽  
Vol 6 (1) ◽  
pp. 46-57
Author(s):  
M. Fang ◽  
S. Fenner ◽  
F. Green ◽  
S. Homer ◽  
Y. Zhang
Keyword(s):  
Lower Bounds ◽  
Quantum Circuit ◽  
Circuit Theory ◽  
Quantum Circuits ◽  
Constant Number ◽  
Constant Depth ◽  
Arbitrary Size ◽  
Single Qubit ◽  
Toffoli Gates ◽  
Circuit Family

We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number $a$ of ancill\ae\ and $n$ input qubits, we give a tradeoff between $a$ and the required depth, that results in a non-constant lower bound for fanout when $a = n^{1-o(1)}$. We also show that, regardless of the number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.

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