Improving Schrödinger Equation Implementations with Gray Code for Adiabatic Quantum Computers

Chang, Chia Cheng; McElvain, Kenneth; Rrapaj, Ermal; Wu, Yudong

doi:10.1103/prxquantum.3.020356

Cited by 4 publications

(6 citation statements)

References 28 publications

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“…The initial state chosen is the transverse one, |Ψ 0 = ⊗ |+ . This wavefunction is an eigenstate of the kinetic term, and it reproduces the continuous adiabatic evolution setup of [20]. In addition, it has the added benefit of being implementable in a single step by simultaneously applying the Hadamard gate to each qubit.…”

Section: Experiments On Universal Quantum Computersmentioning

confidence: 98%

“…The next step is to encode the positions in states of qubits. Following our previous work [20], positions are associated with n-body qubit states in the computational basis, and we identify each basis state's amplitude with the wave function at the corresponding point. Alternatively, one could associate position i with a set of qubits [24,25], but the number of qubits, in that case, would be comparable to the number of classical bits (no storage advantage).…”

Section: Laplacianmentioning

confidence: 99%

“…Gray code is an alternative compact binary encoding of integers 0 to 2 n −1 into n bits. The recursion equation for BRGC encoded Laplacian from [20] can be rewritten as an iterative sum,…”

Section: Brgc Encodingmentioning

confidence: 99%

“…As such, it is the dominating contribution to the Trotter error for any number of qubits. The adiabatic schedule was chosen from [20,40,41], and it suffices to obtain the ground state with t = 10 MeV −1 evolution time. We opted to perform 2000 uniform steps with ∆t = t/2000.…”

Section: Experiments On Universal Quantum Computersmentioning

confidence: 99%

“…The Binary Reflected Gray Code (BRGC) encoding can provide significant simplification for quantum computing as it yields simpler spin Hamlitonians, to be simulated by qubits. For instance, it has been proposed * ermalrrapaj@berkeley.edu for encoding ladder states in d-level systems in gate based quantum computing [18,19], and to map the Schrödinger equation in position space, in any dimension, to the XZ model [20].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

Rrapaj¹,

McElvain²,

Chang³

et al. 2022

Preprint

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We study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time t with this circuit, and spectral norm error , we find the circuit depth required is O(t 2 nAD/ ) with n − 3 auxiliary qubits for a system with 2 n lattice points per dimension D and particle number A; an improvement over binary position encoding which requires an exponential number of n-local operators. Further, under the reasonable assumption that [T, V ] bounds ∆t, with T the kinetic energy and V a non-trivial potential, the cost of even an approximate QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as O (n log(n)) while BRGC scales as O (n), giving a definite advantage to the BRGC implementation.

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Section: Experiments On Universal Quantum Computersmentioning

confidence: 98%

Section: Laplacianmentioning

confidence: 99%

Section: Brgc Encodingmentioning

confidence: 99%

Section: Experiments On Universal Quantum Computersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

Rrapaj¹,

McElvain²,

Chang³

et al. 2022

Preprint

Self Cite

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How to experimentally evaluate the adiabatic condition for quantum annealing

Mori,

Kawabata,

Matsuzaki

2024

Sci Rep

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We propose an experimental method for evaluating the adiabatic condition during quantum annealing (QA), which will be essential for solving practical problems. The adiabatic condition consists of the transition matrix element and the energy gap, and our method simultaneously provides information about these components without diagonalizing the Hamiltonian. The key idea is to measure the power spectrum of a time domain signal by adding an oscillating field during QA, and we can estimate the values of the transition matrix element and energy gap from the measurement output. Our results provides a powerful experimental basis for analyzing the performance of QA.

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Development of optimization method for truss structure by quantum annealing

Honda,

Endo,

Kaji

et al. 2024

Sci Rep

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In this study, we developed a new method of topology optimization for truss structures by quantum annealing. To perform quantum annealing analysis with real variables, representation of real numbers as a sum of random number combinations is employed. The nodal displacement is expressed with binary variables. The Hamiltonian H is formulated on the basis of the elastic strain energy and position energy of a truss structure. It is confirmed that truss deformation analysis is possible by quantum annealing. For the analysis of the optimization method for the truss structure, the cross-sectional area of the truss is expressed with binary variables. The iterative calculation for the changes in displacement and cross-sectional area leads to the optimal structure under the prescribed boundary conditions.

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Improving Schrödinger Equation Implementations with Gray Code for Adiabatic Quantum Computers

Cited by 4 publications

References 28 publications

Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

How to experimentally evaluate the adiabatic condition for quantum annealing

Development of optimization method for truss structure by quantum annealing

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