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Digital quantum simulation of fermionic models with a superconducting circuit

R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I. -C. Hoi, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, E. Solano, John M. Martinis

arXiv:1501.07703v1quant-phcond-mat.mes-hallcond-mat.supr-con

TL;DR

Universal digital simulation of interacting fermions is difficult because fermionic statistics require non-local control. The paper implements Hubbard-model dynamics in a superconducting circuit using Trotterized spin-operator decompositions and optimized tunable CZφ gates. The entangling-gate and two-mode Trotter-step errors are 0.020 and 0.074, respectively, while digital error is negligible for two modes but significant for three modes in the time-dependent model.

Abstract

from arXiv · show

Simulating quantum physics with a device which itself is quantum mechanical, a notion Richard Feynman originated, would be an unparallelled computational resource. However, the universal quantum simulation of fermionic systems is daunting due to their particle statistics, and Feynman left as an open question whether it could be done, because of the need for non-local control. Here, we implement fermionic interactions with digital techniques in a superconducting circuit. Focusing on the Hubbard model, we perform time evolution with constant interactions as well as a dynamic phase transition with up to four fermionic modes encoded in four qubits. The implemented digital approach is universal and allows for the efficient simulation of fermions in arbitrary spatial dimensions. We use in excess of 300 single-qubit and two-qubit gates, and reach global fidelities which are limited by gate errors. This demonstration highlights the feasibility of the digital approach and opens a viable route towards analog-digital quantum simulation of interacting fermions and bosons in large-scale solid state systems.

A. Pulse sequences and gate counts

The supplementary material specifies pulse sequences for single Trotter steps with two, three, and four fermionic modes.

A. Pulse sequences and gate counts

B. Initialization

Initialization sequences are given for the three- and four-mode simulations, alongside explicit input states for the two- and four-mode cases.

B. Initialization

II. QUANTUM PROCESS TOMOGRAPHY

Quantum process tomography reconstructs process matrices from a complete set of prepared and measured two-qubit states, with experimentally obtained component processes combined to evaluate their product.

II. QUANTUM PROCESS TOMOGRAPHY

  • The χ matrix is reconstructed from 16 output density matrices generated using I, X/2, Y/2, and X preparations and measurements.Quadratic maximum-likelihood estimation constrains the result to be Hermitian, trace-preserving, and positive semidefinite.Source: §II · p. 6
  • Tomography on a zero-time idle suppresses non-idealities from measurement and state preparation.Source: §II · p. 6
  • The experimentally determined component process matrices are combined to compute the process U2U1.Source: §II · p. 7

III. RANDOMIZED BENCHMARKING OF exp(−i π 4 σz ⊗σz) AND THE TWO-MODE TROTTER STEP

Randomized benchmarking evaluates the entangling gate and the two-mode Trotter step using Clifford-compatible representative angles and sequence-fidelity decay.

III. RANDOMIZED BENCHMARKING OF exp(−i π 4 σz ⊗σz) AND THE TWO-MODE TROTTER STEP

  • Representative Clifford-compatible angles are φ = π/2 and φxx = φyy = φzz = π/2.Source: §III · p. 7
  • Interleaved randomized benchmarking links the additional sequence-fidelity decrease to the process fidelity of the interleaved gate.Source: §III · p. 7
  • The exp(−i π 4 σz ⊗σz) gate and the two-mode Trotter step have errors of 0.020 and 0.074, respectively.These values are consistent with estimates obtained by adding individual gate errors.Source: §III · p. 7

IV. DIGITAL ERROR

The Trotter expansion introduces discretization errors whose size depends on the model and number of Trotter steps. Errors are negligible for the two-mode cases but become significant for larger or more coarsely digitized simulations.

  • The two-mode time-independent simulation has zero digital error, whereas three- and four-mode simulations exhibit digital-error deviations.Source: §IV · p. 7
  • The time-dependent simulation has negligible digital error for two modes but significant error for three modes.The three-mode error arises from approximating a larger Hamiltonian with only one Trotter step.Source: §IV · p. 7

V. MINIMIZING LEAKAGE OF THE CZφ GATE

The tunable CZφ gate can leak population when large phases require approaching the |ee⟩–|gf⟩ avoided crossing. Lengthening the gate and optimizing its waveform suppresses this leakage.

  • Large CZφ phases require close approach to the |ee⟩–|gf⟩ avoided crossing, which induces state leakage.The gate uses an adiabatic frequency trajectory to tune the qubit toward the avoided crossing.Source: §V · p. 7
  • Figure S5 compares ideal evolution, exact digital solutions, and fidelity for time-dependent simulations with two modes using two Trotter steps and three modes using one step.Source: Figure S5 · p. 8
  • For phases above 4.0 rads, increasing the CZφ gate length from 40 ns to 55 ns was insufficient to eliminate considerable leakage.Source: §V · p. 8
  • Nelder-Mead optimization of waveform parameters using leaked-state population as the fitness metric significantly suppresses leakage.The optimization took approximately one minute in real time.Source: §V · p. 8

VI. ASYMMETRIC HUBBARD MODEL

The asymmetric Hubbard model extends the Hubbard framework to anisotropic fermionic systems with potentially unequal hopping terms. The paper maps its fermionic operators to Pauli operators so the resulting interactions can be implemented digitally in a superconducting circuit.

  • The asymmetric Hubbard model describes anisotropic fermionic systems with two fermionic species on two lattice sites.Source: §VI · p. 8
  • The Jordan-Wigner transformation maps fermionic operators to tensor products of Pauli matrices that can be simulated in the superconducting circuit.Source: §VI · p. 8
  • The transformed four-qubit Hamiltonian contains XX, YY, and ZZ-type interactions together with single-qubit terms.Source: §VI · pp. 8–9
  • The resulting interactions are implemented digitally using single-qubit gates and tunable CZφ gates.Source: §VI · p. 9

A. Gate decomposition

The gate decomposition digitizes each Hamiltonian term through Trotter steps built from ZZ interactions and single-qubit rotations. Gate ordering balances Trotter-error behavior against experimental gate-count optimization, with a particular asymmetric case enabling further cancellations.

A. Gate decomposition

  • Each Trotter step sequentially applies the evolution operators of the Hamiltonian’s summands over time intervals of length t/n.Source: A. Gate decomposition · p. 9
  • XX and YY interactions are converted into ZZ interactions using single-qubit rotations around the y and x axes, respectively.Source: A. Gate decomposition · p. 9
  • Gate ordering changes results when the number of Trotter steps is limited to approximately n ≤ 10, while the global digital error scales as 1/n.Source: A. Gate decomposition · p. 9
  • The experimental ordering prioritizes gate-count reduction and removal of global phases, even though grouping commuting terms can improve Trotter fidelity.Source: A. Gate decomposition · p. 9

B. Particular case of the model

C. Digital quantum simulation of the model

The four-mode experimental system is framed as a digital simulation of the fermionic asymmetric Hubbard model, using optimized CZφ gates within Trotter steps. The numerical treatment relates simulation parameters to gate phases and assumes ℏ=1.

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