Digital quantum simulation of fermionic models with a superconducting circuit

Abstract

One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions.

Figure 1. Model and device.

(a) Hubbard model picture with two sites and four modes, with hopping strength V and on-site interactions U. The creation of one excitation from the groundstate is shown for each mode. (b) Optical micrograph of the device. The scale bar (bottom left) denotes 200 μm. The coloured cross-shaped structures are the used Xmon transmon qubits. The construction of the fermionic operators for four modes is shown on the right. Colours highlight the corresponding sites, qubits and operators.

Figure 2. Gate construction.

(a) Construction of the gate from single-qubit rotations and the tunable CZ_φ-entangling gate. To enable small and negative angles, we include π pulses around the x axis (A=X) or y axis (A=Y). The unitary diagonals are (1 e^iφ e^iφ 1). (b) Tunable CZ_φ gate, implemented by moving |ee〉 (red) close to |gf〉 (blue). Coupling strength is g/2π=14 MHz, pulse length is 55 ns, and typically Δ/2π=0.7 GHz when idling. (c) Measured versus desired phase of the full sequence, determined using quantum state tomography.

Figure 3. Quantum process tomography of operator anticommutation.

The process matrices are shown for the non-trivial Hermitian terms of the anticommutation relations. (a) Process matrix of the unitary . (b) Process matrix of the unitary . (c) The sequence of both processes, , yields the identity. The significant matrix elements, red for the real and blue for the imaginary elements, are close to the ideal (transparent).

Figure 4. Simulation of two fermionic modes.

(a) Construction of the two-mode Trotter step, showing the separate terms of the Hamiltonian (equation (2)). See Supplementary Note 1 for the pulse sequence and gate count. (b) Occupation of the modes versus simulated time for n=1,...,8 steps. Colour coding denotes the state probabilities. Input state is , and V=U=1. The ideal dependence is shown in the bottom right. The final simulation time is T=5. (c) The end-state fidelity decreases with step by 0.054, following a linear trend.

Figure 5. Fermionic models with three and four modes.

(a) Three-mode Trotter step, with the Trotter step pulse sequence in b. The Trotter step consists of 12 entangling gates and 87 single-qubit gates (see text). The interaction is highlighted (dashed). The amplitudes of the rotations are controlled by the values of V and U: , and . (c) Simulation results for three modes with and without on-site interaction. Full symbols: experiment. Open symbols: ideal digitized. Black symbols: population of other states. Input state is , and V=1. (d) Construction of the four-mode Trotter step. The amplitudes of the rotations are: , and . (e) Four-mode simulation results for V₁=V₂=1, U₂₃=1 and U₁₄=0. Input state is . (f) Fidelities versus Trotter step for the three-mode simulation (dots) and the four-mode simulation (triangles).

Figure 6. Simulations with time-varying interactions.

(a) The system is changed from an insulating state (denoted by the blue background) to a conducting phase (denoted by a red background), by ramping the hopping term V from zero to one. Solid line: U, dashed line: V. Inset shows the choice of digitization on the ramp for the two-mode simulation. (b) Two-mode simulation showing dynamic behaviour starting at the onset of the V ramp. Dashed lines denote the ideal digitized evolution. (c) Three-mode simulation, showing non-trivial dynamics when the hopping term is non-zero. Dashed lines denote the ideal digitized evolution. Black symbols indicate the population of other states. (d) Simulation fidelities.