scholarly journals Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 322
Author(s):  
Ewout van den Berg ◽  
Kristan Temme
Keyword(s):  
Quantum Chemistry ◽  
Time Evolution ◽  
Circuit Complexity ◽  
Practical Interest ◽  
Quantum Circuits ◽  
Circuit Optimization ◽  
Simultaneous Diagonalization ◽  
Hamiltonian Simulation ◽  
Practical Algorithm ◽  
Circuit Depth

Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of CNOT operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.

scholarly journals Exponentially faster implementations of Select(H) for fermionic Hamiltonians

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 380
Author(s):  
Kianna Wan
Keyword(s):  
Quantum Chemistry ◽  
General Framework ◽  
State Of The Art ◽  
Quantum Algorithms ◽  
Condensed Matter ◽  
Quantum Circuits ◽  
Constant Independent ◽  
Wigner Transform ◽  
Gate Count ◽  
Hamiltonian Simulation

We present a simple but general framework for constructing quantum circuits that implement the multiply-controlled unitary Select(H):=∑ℓ|ℓ⟩⟨ℓ|⊗Hℓ, where H=∑ℓHℓ is the Jordan-Wigner transform of an arbitrary second-quantised fermionic Hamiltonian. Select(H) is one of the main subroutines of several quantum algorithms, including state-of-the-art techniques for Hamiltonian simulation. If each term in the second-quantised Hamiltonian involves at most k spin-orbitals and k is a constant independent of the total number of spin-orbitals n (as is the case for the majority of quantum chemistry and condensed matter models considered in the literature, for which k is typically 2 or 4), our implementation of Select(H) requires no ancilla qubits and uses O(n) Clifford+T gates, with the Clifford gates applied in O(log2n) layers and the T gates in O(logn) layers. This achieves an exponential improvement in both Clifford- and T-depth over previous work, while maintaining linear gate count and reducing the number of ancillae to zero.

scholarly journals Quantum circuit optimization using quantum Karnaugh map

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
J.-H. Bae ◽  
Paul M. Alsing ◽  
Doyeol Ahn ◽  
Warner A. Miller
Keyword(s):  
Quantum Computation ◽  
Quantum Algorithms ◽  
Optimization Technique ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Circuit Optimization ◽  
General Technique ◽  
Small Quantum

Abstract Every quantum algorithm is represented by set of quantum circuits. Any optimization scheme for a quantum algorithm and quantum computation is very important especially in the arena of quantum computation with limited number of qubit resources. Major obstacle to this goal is the large number of elemental quantum gates to build even small quantum circuits. Here, we propose and demonstrate a general technique that significantly reduces the number of elemental gates to build quantum circuits. This is impactful for the design of quantum circuits, and we show below this could reduce the number of gates by 60% and 46% for the four- and five-qubit Toffoli gates, two key quantum circuits, respectively, as compared with simplest known decomposition. Reduced circuit complexity often goes hand-in-hand with higher efficiency and bandwidth. The quantum circuit optimization technique proposed in this work would provide a significant step forward in the optimization of quantum circuits and quantum algorithms, and has the potential for wider application in quantum computation.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

scholarly journals Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

2022 ◽  
Vol 3 (1) ◽  
pp. 1-37
Author(s):  
Almudena Carrera Vazquez ◽  
Ralf Hiptmair ◽  
Stefan Woerner
Keyword(s):  
Necessary Conditions ◽  
Linear Equations ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Richardson Extrapolation ◽  
Classical Simulation ◽  
Hamiltonian Simulation ◽  
Systems Of Linear Equations ◽  
Amplitude Amplification ◽  
Theoretical Results

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

scholarly journals Energy-participation quantization of Josephson circuits

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Zlatko K. Minev ◽  
Zaki Leghtas ◽  
Shantanu O. Mundhada ◽  
Lysander Christakis ◽  
Ioan M. Pop ◽  
...  
Keyword(s):  
Circuit Complexity ◽  
Quantum Circuits ◽  
Nonlinear Interactions ◽  
Electromagnetic Mode ◽  
System Quantum ◽  
Josephson Circuits ◽  
Nonlinear Devices ◽  
Hamiltonian Parameters ◽  
Mode Energy ◽  
Multi Mode

AbstractSuperconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions, are a leading platform for emerging quantum technologies. Increasing circuit complexity further requires efficient methods for the calculation and optimization of the spectrum, nonlinear interactions, and dissipation in multi-mode distributed quantum circuits. Here we present a method based on the energy-participation ratio (EPR) of a dissipative or nonlinear element in an electromagnetic mode. The EPR, a number between zero and one, quantifies how much of the mode energy is stored in each element. The EPRs obey universal constraints and are calculated from one electromagnetic-eigenmode simulation. They lead directly to the system quantum Hamiltonian and dissipative parameters. The method provides an intuitive and simple-to-use tool to quantize multi-junction circuits. We experimentally tested this method on a variety of Josephson circuits and demonstrated agreement within several percents for nonlinear couplings and modal Hamiltonian parameters, spanning five orders of magnitude in energy, across a dozen samples.

scholarly journals QUANTUM ORACLES IN TERMS OF UNIVERSAL GATE SET

2011 ◽  
Vol 09 (06) ◽  
pp. 1363-1381 ◽  
Author(s):  
YUJI TANAKA ◽  
TSUBASA ICHIKAWA ◽  
MASAHITO TADA-UMEZAKI ◽  
YUKIHIRO OTA ◽  
MIKIO NAKAHARA
Keyword(s):  
Arbitrary Number ◽  
Search Algorithm ◽  
Circuit Complexity ◽  
Database Search ◽  
Quantum Circuits ◽  
Systematic Construction ◽  
Universal Gate

We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.

scholarly journals Revisiting the simulation of quantum Turing machines by quantum circuits

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180767 ◽  
Author(s):  
Abel Molina ◽  
John Watrous
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Circuit Complexity ◽  
Simulation Method ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Turing Machines ◽  
Generated Family ◽  
Quantum Turing Machine

Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.

scholarly journals Tracking the excited-state time evolution of the visual pigment with multiconfigurational quantum chemistry

2007 ◽  
Vol 104 (19) ◽  
pp. 7764-7769 ◽  
Author(s):  
L. M. Frutos ◽  
T. Andruniow ◽  
F. Santoro ◽  
N. Ferre ◽  
M. Olivucci
Keyword(s):  
Excited State ◽  
Quantum Chemistry ◽  
Time Evolution ◽  
Visual Pigment ◽  
State Time

scholarly journals Improved Hamiltonian simulation via a truncated Taylor series and corrections

2017 ◽  
Vol 17 (7&8) ◽  
pp. 623-635
Author(s):  
Leonardo Novo ◽  
Dominic Berry
Keyword(s):  
Quantum Chemistry ◽  
Taylor Series ◽  
Evolution Operator ◽  
Quantum Algorithm ◽  
Simulation Method ◽  
Quantum Simulation ◽  
Wide Range ◽  
Simulation Based ◽  
Extra Step ◽  
Hamiltonian Simulation

We describe an improved version of the quantum algorithm for Hamiltonian simulation based on the implementation of a truncated Taylor series of the evolution operator. The idea is to add an extra step to the previously known algorithm which implements an operator that corrects the weightings of the Taylor series. This way, the desired accuracy is achieved with an improvement in the overall complexity of the algorithm. This quantum simulation method is applicable to a wide range of Hamiltonians of interest, including to quantum chemistry problems.

scholarly journals Dimensional Expressivity Analysis of Parametric Quantum Circuits

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 422
Author(s):  
Lena Funcke ◽  
Tobias Hartung ◽  
Karl Jansen ◽  
Stefan Kühn ◽  
Paolo Stornati
Keyword(s):  
Crucial Role ◽  
Quantum Algorithms ◽  
Solution Space ◽  
Quantum Circuits ◽  
Classical Approach ◽  
Circuit Layout ◽  
Minimum Number ◽  
Circuit Depth ◽  
Proof Of Principle

Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

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