Fixed Depth Hamiltonian Simulation via Cartan Decomposition

Kökcü, Efekan; Steckmann, Thomas; Wang, Yan; Freericks, J. K.; Dumitrescu, Eugene; Kemper, A. F.

doi:10.1103/physrevlett.129.070501

Cited by 24 publications

(7 citation statements)

References 50 publications

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“…The advantage of NC can be seen not only for electronic Hamiltonians. Consider the following model Hamiltonian, which is an extension of the Heisenberg spin Hamiltonian .25ex2ex H = ̂ ∑ i = 1 2 n − 1 false( a i x̂ i x̂ i + 1 + b i ŷ i ŷ i + 1 + c i ẑ i ẑ i + 1 false) infix+ ∑ j = 1 2 n d j ẑ j …”

Section: Resultsmentioning

confidence: 99%

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras

Patel,

Yen,

Izmaylov

2024

J. Phys. Chem. A

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Exactly solvable Hamiltonians that can be diagonalized by using relatively simple unitary transformations are of great use in quantum computing. They can be employed for the decomposition of interacting Hamiltonians either in Trotter-Suzuki approximations of the evolution operator for the quantum phase estimation algorithm or in the quantum measurement problem for the variational quantum eigensolver. One of the typical forms of exactly solvable Hamiltonians is a linear combination of operators forming a modestly sized Lie algebra. Very frequently, such linear combinations represent noninteracting Hamiltonians and thus are of limited interest for describing interacting cases. Here, we propose an extension in which the coefficients in these combinations are substituted by polynomials of the Lie algebra symmetries. This substitution results in a more general class of solvable Hamiltonians, and for qubit algebras, it is related to the recently proposed noncontextual Pauli Hamiltonians. In fermionic problems, this substitution leads to Hamiltonians with eigenstates that are single Slater determinants but with different sets of singleparticle states for different eigenstates. The new class of solvable Hamiltonians can be measured efficiently using quantum circuits with gates that depend on the result of a midcircuit measurement of the symmetries.

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Section: Resultsmentioning

confidence: 99%

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras

Patel,

Yen,

Izmaylov

2024

J. Phys. Chem. A

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“…Naturally, one could attribute the large effectiveness of the circuit optimization to the integrable property of the TFIM. In this sense, our method could be regarded as numerical equivalent of circuit compression based on the integrable structure manifest in the Yang-Baxter equation, as studied in [28][29][30][31]. Surprisingly, however, the optimization results remain qualitatively unaffected by turning on an integrability-breaking longitudinal field, i.e.…”

Section: Ising Model On a One-dimensional Latticementioning

confidence: 99%

Riemannian quantum circuit optimization for Hamiltonian simulation

Kotil,

Banerjee,

Huang

et al. 2024

J. Phys. A: Math. Theor.

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Hamiltonian simulation, i.e., simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation (TEBD) algorithm.

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“…as can be seen in figure 3. These gates can be synthetized for SU(2 N ) operations with arbitrary number of qubits by means of Cartan decomposition [44][45][46][47], and other alternative methods [48,49]. Yet, the complexity and depth of the quantum circuits increases exponentially with the number of qubits producing fast fidelity decays.…”

Section: Quantum Circuitmentioning

confidence: 99%

Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Kunold

2024

Phys. Scr.

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Based oh the properties of Lie algebras,in this work we develop a general frameworkto linearize the von Neumann equationrendering it in a suitable formfor quantum simulations.Departing from the conventionalmethod of expanding the densitymatrix in the Liouville space formed by matricesunit we express the von Neumannequation in terms of Pauli strings.This provides several advantagesrelated to the quantum tomography of the densitymatrix and the formulation ofthe unitary gates that generatethe time evolution.The use of Pauli strings facilitates thequantum tomography of the density matrixwhose elements are purely real.As for any other basis of Hermitian matrices,this eliminates the need to calculatethe phase of the complex entries of thedensity matrix.This approach also enables to expressthe evolution operator as a sequenceof commuting Hamiltonian gatesof Pauli strings that can readilybe synthetized using Clifford gates.Additionally, the fact that these gates commutewith each other alongwith the unique properties of the algebra formed by Paulistrings allows to avoid the use of Trotterizationhence considerably reducing the circuit depth.The algorithm is demonstrated for threeHamiltonians using the IBMnoisy quantum circuitsimulator.

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Fixed Depth Hamiltonian Simulation via Cartan Decomposition

Cited by 24 publications

References 50 publications

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras

Riemannian quantum circuit optimization for Hamiltonian simulation

Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

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