scholarly journals Corrected quantum walk for optimal Hamiltonian simulation

2016 ◽  
Vol 16 (15&16) ◽  
pp. 1295-1317
Author(s):  
Dominic Berry ◽  
Leonardo Novo
Keyword(s):  
Lower Bound ◽  
Quantum Chemistry ◽  
Taylor Series ◽  
Quantum Computer ◽  
Quantum Walk ◽  
Query Complexity ◽  
Hamiltonian Simulation ◽  
Hamiltonian Evolution

We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to doublelogarithmic factors for all parameter regimes. The scaling of the query complexity is given. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.

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 An Exponential Separation Between MA and AM Proofs of Proximity

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Tom Gur ◽  
Yang P. Liu ◽  
Ron D. Rothblum
Keyword(s):  
Quantum Information ◽  
Lower Bound ◽  
Recent Result ◽  
Query Complexity ◽  
Random String ◽  
Natural Property ◽  
Public And Private ◽  
Alternate Proof ◽  
Interactive Proofs ◽  
Exponential Separation

AbstractInteractive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as -proofs of Proximity (), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as -proofs of Proximity (), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both s and s is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of s and s. Specifically, we exhibit an explicit and natural property $$\Pi$$ Π that admits an with complexity $$O(\log n)$$ O ( log n ) , whereas any for $$\Pi$$ Π has complexity $$\tilde{\Omega}(n^{1/4})$$ Ω ~ ( n 1 / 4 ) , where n denotes the length of the input in bits. Our lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a $$\Omega(n^{1/6})$$ Ω ( n 1 / 6 ) lower bound for the same property $$\Pi$$ Π .Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that s can only be quadratically stronger than s. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.

scholarly journals Qubit Coupled Cluster Method: A Systematic Approach to Quantum Chemistry on a Quantum Computer

2018 ◽  
Vol 14 (12) ◽  
pp. 6317-6326 ◽  
Author(s):  
Ilya G. Ryabinkin ◽  
Tzu-Ching Yen ◽  
Scott N. Genin ◽  
Artur F. Izmaylov
Keyword(s):  
Quantum Chemistry ◽  
Systematic Approach ◽  
Quantum Computer ◽  
Cluster Method ◽  
Coupled Cluster ◽  
Coupled Cluster Method

Quantum chemistry beyond Born-Oppenheimer approximation on a quantum computer: A simulated phase estimation study

2016 ◽  
Vol 116 (18) ◽  
pp. 1328-1336 ◽  
Author(s):  
Libor Veis ◽  
Jakub Višňák ◽  
Hiroaki Nishizawa ◽  
Hiromi Nakai ◽  
Jiří Pittner
Keyword(s):  
Quantum Chemistry ◽  
Quantum Computer ◽  
Phase Estimation ◽  
Oppenheimer Approximation

scholarly journals QUANTUM TIMING AND SYNCHRONIZATION PROBLEMS

2004 ◽  
Vol 18 (04n05) ◽  
pp. 623-631 ◽  
Author(s):  
DIEGO DE FALCO ◽  
DARIO TAMASCELLI
Keyword(s):  
Continuous Time ◽  
Quantum Computer ◽  
Linear Chain ◽  
Quantum Walk ◽  
Storage Space ◽  
Input Output ◽  
Output Register ◽  
Additional Amount ◽  
Time Quantum ◽  
Synchronization Problems

Feynman's model of a quantum computer provides an example of a continuous-time quantum walk. Its clocking mechanism is an excitation of a basically linear chain of spins with occasional controlled jumps which allow for motion on a planar graph. The spreading of the wave packet poses limitations on the probability of ever completing the s elementary steps of a computation: an additional amount of storage space δ is needed in order to achieve an assigned completion probability. In this note we study the END instruction, viewed as a measurement of the position of the clocking excitation: a π-pulse indefinitely freezes the contents of the input/output register, with a probability depending only on the ratio δ/s.

scholarly journals A random walk approach to quantum algorithms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3407-3422 ◽  
Author(s):  
Vivien M Kendon
Keyword(s):  
Random Walk ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Random Walk ◽  
Small Quantum ◽  
Starting Point ◽  
Quantum Version ◽  
Speed Up

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.

scholarly journals A Full Quantum Eigensolver for Quantum Chemistry Simulations

Research ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Shijie Wei ◽  
Hang Li ◽  
GuiLu Long
Keyword(s):  
Quantum Computing ◽  
Quantum Chemistry ◽  
Gradient Descent ◽  
Materials Science ◽  
Hybrid Methods ◽  
Quantum Computer ◽  
Rapid Development ◽  
System Size ◽  
Near Term ◽  
Full Quantum

Quantum simulation of quantum chemistry is one of the most compelling applications of quantum computing. It is of particular importance in areas ranging from materials science, biochemistry, and condensed matter physics. Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the molecular ground energies and electronic structures using quantum gradient descent. Compared to existing classical-quantum hybrid methods such as variational quantum eigensolver (VQE), our method removes the classical optimizer and performs all the calculations on a quantum computer with faster convergence. The gradient descent iteration depth has a favorable complexity that is logarithmically dependent on the system size and inverse of the precision. Moreover, the FQE can be further simplified by exploiting a perturbation theory for the calculations of intermediate matrix elements and obtaining results with a precision that satisfies the requirement of chemistry application. The full quantum eigensolver can be implemented on a near-term quantum computer. With the rapid development of quantum computing hardware, the FQE provides an efficient and powerful tool to solve quantum chemistry problems.

scholarly journals The quantum query complexity of AC0

2012 ◽  
Vol 12 (7&8) ◽  
pp. 670-676
Author(s):  
Paul Beame ◽  
Widad Machmouchi
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Input Function ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Quantum Query

We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from $[2n-2]$ to $[n]$ is surjective. These results yield a nearly linear $\Omega(n/\log n)$ lower bound on the quantum query complexity of $\cl{AC}^0$. The best previous lower bound known for any $\cl{AC^0}$ function was the $\Omega ((n/\log n)^{2/3})$ bound given by Aaronson and Shi's $\Omega(n^{2/3})$ lower bound for the element distinctness problem.

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