Improved Fermionic Scattering for the NISQ Era

TL;DR

A scattering state preparation method is proposed that approximates the fermionic wave packets by localizing them in space, reducing circuit depth by nearly half, while also preserving fermionic anti-commutation relations.

Abstract

In the era of noisy intermediate scale quantum (NISQ) hardware, digital quantum computers are limited to shallow circuits on the order of a thousand layers due to system noise and qubit decoherence. Thus, every step of a simulation must be as efficient as possible. Modifying the recent Givens Rotation state preparation by Chai et al and ladder operator block encoding method by Simon et al, we propose a scattering state preparation method that approximates the fermionic wave packets by localizing them in space, reducing circuit depth by nearly half, while also preserving fermionic anti-commutation relations. Using MPS simulations, we show that these approximated wave packets approach the exact wave packets in weakly interacting critical theories; and then show its immediate application on modern day hardware with IonQ's Forte 1 machine.

Figures (5)

• Figure 1: The Trotterized time evolution operator for the modified Ising model. The boundary term is assumed but not shown. All Pauli-rotation gates are defined as $RP(\theta)=\exp(-i\hat{P}\;\theta/2)$. The layers $H$RZZ$H$ correspond to RXX layers, where $H$ is the Hadamard gate.
• Figure 2: (a) Full scattering state preparation circuit in an 8-site space. The ground state is prepared separately with VQE, then two narrow wave packets are built in subspaces 0-3 and 4-7. The string of Pauli-$z$ gates restore Fermionic anti-commutation relations for the second wave packet. (b) Decomposition of $\mathcal{V}^\dagger(\beta,\theta)$ in terms of single and two qubit operations. (c) Equivalent unitary operation for $\sigma^-$ using a control and ancilla qubit. (d) Decomposition of the two qubit Givens rotation operation in (b). The CNOT depth of the circuit in (a) (excluding ground state preparation) is $18$.
• Figure 3: MPS Evolution of (a) site occupations $\langle N_j(t)\rangle$ and (b) bipartite entanglement entropy of regions $(0,j+1)$ to the rest of the system $(j+2,N-2)$ for a scattering state with initial mean positions $x_A=3,\;x_B=11$, momenta $k=\pm7\pi/16$, and width $\sigma=3/2$ for $J=0.4,\;h=1,\;g=0.01,\;\Delta t=0.1$.
• Figure 4: Initial state prepared on IonQ Forte 1 for an 8-site scattering state with initial positions $x=1,5$, momenta $k=\frac{3\pi}{8}$, and width $\sigma=3/2$.
• Figure 5: Example qubit map on IBM's heavy hexagonal topology. The blue circles represent system qubits, the white non-system qubits, and the red and green as control and ancilla qubits respectively that are used to implement $\sigma^-$. In this case, the system size is 28.