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 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.

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 Improved Dynamic Programming Algorithm for Bitonic TSP

2013 ◽  
Vol 347-350 ◽  
pp. 3094-3098 ◽  
Author(s):  
Jian Li
Keyword(s):  
Dynamic Programming ◽  
Time Complexity ◽  
Dynamic Programming Algorithm ◽  
Space Complexity ◽  
Programming Algorithm ◽  
Time And Space ◽  
Space Requirement ◽  
Classical Algorithm ◽  
Computing Speed ◽  
Improved Dynamic Programming

This paper puts forward an improved dynamic programming algorithm for bitonic TSP and it proves to be correct. Divide the whole loop into right-and-left parts through analyzing the key point connecting to the last one directly; then construct a new optimal sub-structure and recursion. The time complexity of the new algorithm is O(n2) and the space complexity is O(n); while both the time and space complexities of the classical algorithm are O(n2). Experiment results showed that the new algorithm not only reduces the space requirement greatly but also increases the computing speed by 2-3 times compared with the classical algorithm.

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 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 Efficient circuits for exact-universal computation with qudits

2006 ◽  
Vol 6 (4&5) ◽  
pp. 436-454
Author(s):  
G.K. Brennen ◽  
S.S. Bullock ◽  
D.P. O'Leary
Keyword(s):  
Lower Bound ◽  
Matrix Decomposition ◽  
Efficient Implementation ◽  
The State ◽  
Quantum Circuits ◽  
Asymptotically Optimal ◽  
Original Algorithm ◽  
Universal Computation

This paper concerns the efficient implementation of quantum circuits for qudits. We show that controlled two-qudit gates can be implemented without ancillas and prove that the gate library containing arbitrary local unitaries and one two-qudit gate, $\CINC$, is exact-universal. A recent paper [S.Bullock, D.O'Leary, and G.K. Brennen, Phys. Rev. Lett. 94, 230502 (2005)] describes quantum circuits for qudits which require O(d^n) two-qudit gates for state synthesis and O(d^{2n}) two-qudit gates for unitary synthesis, matching the respective lower bound complexities. In this work, we present the state-synthesis circuit in much greater detail and prove that it is correct. Also, the (n-2)/(d-2) ancillas required in the original algorithm may be removed without changing the asymptotics. Further, we present a new algorithm for unitary synthesis, inspired by the QR matrix decomposition, which is also asymptotically optimal.

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.

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