scholarly journals Resource-Optimal Single-Qubit Quantum Circuits

2012 ◽  
Vol 109 (19) ◽  
Author(s):  
Alex Bocharov ◽  
Krysta M. Svore
Keyword(s):  
Quantum Circuits ◽  
Single Qubit

scholarly journals DIAGONAL-UNITARY 2-DESIGN AND THEIR IMPLEMENTATIONS BY QUANTUM CIRCUITS

2013 ◽  
Vol 11 (07) ◽  
pp. 1350062 ◽  
Author(s):  
YOSHIFUMI NAKATA ◽  
MIO MURAO
Keyword(s):  
Algebraic Structure ◽  
Quantum Circuits ◽  
Random Choice ◽  
Unitary Matrices ◽  
Single Qubit ◽  
Classical Procedure ◽  
Random States ◽  
Computational Basis

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.

An algorithm for the T-countAn algorithm for the T-count

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1261-1276 ◽  
Author(s):  
David Gosset ◽  
Vadym Kliuchnikov ◽  
Michele Mosca ◽  
Vincent Russo
Keyword(s):  
Simple Expression ◽  
Quantum Circuits ◽  
Time And Space ◽  
Single Qubit ◽  
Universal Computation ◽  
Classical Algorithm ◽  
Fredkin Gate

We consider quantum circuits composed of Clifford and $T$ gates. In this context the $T$ gate has a special status since it confers universal computation when added to the (classically simulable) Clifford gates. However it can be very expensive to implement fault-tolerantly. We therefore view this gate as a resource which should be used only when necessary. Given an $n$-qubit unitary $U$ we are interested in computing a circuit that implements it using the minimum possible number of $T$ gates (called the $T$-count of $U$). A related task is to decide if the $T$-count of $U$ is less than or equal to $m$; we consider this problem as a function of $N=2^n$ and $m$. We provide a classical algorithm which solves it using time and space both upper bounded as $\mathcal{O}(N^m \text{poly}(m,N))$. We implemented our algorithm and used it to show that any Clifford+T circuit for the Toffoli or the Fredkin gate requires at least 7 $T$ gates. This implies that the known 7 $T$ gate circuits for these gates are $T$-optimal. We also provide a simple expression for the $T$-count of single-qubit unitaries.

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.

scholarly journals Improved upper bounds on the stabilizer rank of magic states

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 606
Author(s):  
Hammam Qassim ◽  
Hakop Pashayan ◽  
David Gosset
Keyword(s):  
Relative Error ◽  
Upper Bound ◽  
Linear Codes ◽  
Upper Bounds ◽  
Quantum Circuits ◽  
Symmetric Product ◽  
Simulation Algorithm ◽  
Single Qubit ◽  
Classical Algorithm ◽  
Stabilizer States

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=2−1(|0⟩+eiπ/4|1⟩) in the limit of large m. In particular, we show that |T⟩⊗m can be exactly expressed as a superposition of at most O(2αm) stabilizer states, where α≤0.3963, improving on the best previously known bound α≤0.463. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n-qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime poly(n,m)2αm. We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩⊗m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z-rotation gate with runtime poly(n,m)2m/2. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

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 Quantum Circuits for Sparse Isometries

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 412
Author(s):  
Emanuel Malvetti ◽  
Raban Iten ◽  
Roger Colbeck
Keyword(s):  
Quantum Computation ◽  
Quantum Circuits ◽  
State Preparation ◽  
Single Qubit

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

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.

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1149-1164
Author(s):  
Yasuhiro Takahashi ◽  
Takeshi Yamazaki ◽  
Kazuyuki Tanaka
Keyword(s):  
Probability Distribution ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Constant Depth ◽  
Toffoli Gate ◽  
Single Qubit ◽  
Polynomial Size ◽  
Toffoli Gates ◽  
Output Probability ◽  
Universal Gates

We study the classical simulatability of constant-depth polynomial-size quantum circuits followed by only one single-qubit measurement, where the circuits consist of universal gates on at most two qubits and additional gates on an unbounded number of qubits. First, we consider unbounded Toffoli gates as additional gates and deal with the weak simulation, i.e., sampling the output probability distribution. We show that there exists a constant-depth quantum circuit with only one unbounded Toffoli gate that is not weakly simulatable, unless $\bqp \subseteq \postbpp \cap \am$. Then, we consider unbounded fan-out gates as additional gates and deal with the strong simulation, i.e., computing the output probability. We show that there exists a constant-depth quantum circuit with only two unbounded fan-out gates that is not strongly simulatable, unless $\p = \pp$. These results are in contrast to the fact that any constant-depth quantum circuit without additional gates on an unbounded number of qubits is strongly and weakly simulatable.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

scholarly journals Andreev Probe of Persistent Current States in Superconducting Quantum Circuits

2005 ◽  
Vol 95 (14) ◽  
Author(s):  
V. T. Petrashov ◽  
K. G. Chua ◽  
K. M. Marshall ◽  
R. Sh. Shaikhaidarov ◽  
J. T. Nicholls
Keyword(s):  
Persistent Current ◽  
Quantum Circuits ◽  
Superconducting Quantum Circuits
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