On the stability to noise of fermion-to-qubit mappings

Abstract

Quantum simulations before fault tolerance suffer from the intrinsic noise present in quantum computers. In this regime, extracting meaningful results greatly benefits from stability against that noise. This stability, defined as an error in observables that is independent of the system's size, is expected in local systems under local noise. In fermionic systems, the encoding of the fermionic degrees of freedom into qubits can introduce non-locality, making stability more delicate. Here, we investigate the stability to noise of fermion-to-qubit mappings. We consider noisy quantum circuits in $D$ dimensions modeled by alternating layers of local unitaries and general, single-qubit Pauli noise. We show that, when using local fermionic encodings, expectation values of quadratic fermionic observables are stable to noise in states with spatially decaying correlations: a power-law decay with exponent $μ>D$ is sufficient for stability. By contrast, we show that this stability cannot be achieved by non-local encodings such as Jordan-Wigner in $2D$, or quasi-local ones such as the Bravyi-Kitaev transform. Our findings formalize the intuition that decaying correlations of the physical systems under study provide protection against noise for local fermionic encodings, and help inform design principles in near-term quantum simulations.

Figures (3)

• Figure 1: Response to noise of a $1D$ ground state with Fermi surface. (a) Expectation values of the momentum occupation with momenta $q_0=2\pi /N$ and Fermi momentum $k_F$, as a function of the system size with a fixed noise rate $p=10^{-2}$, and half-filling condition $(k_F=\pi/2)$. As predicted, the momentum expectation value at the Fermi surface rapidly degrades with increasing system size, signalling fragility to noise. On the other hand, the measurement of the occupation at momentum $q_0$ is stable to noise and does not deteriorate for large system sizes. (b) Representation of the sensitivity to noise of the noisy expectation value $\langle n_k\rangle_{\mathrm{noisy}}$ for a fixed system size $N=100$. The sensitivity to noise for a fixed momentum $k$ is estimated as the slope $| \langle n_k\rangle_{\mathrm{noisy}}-\langle n_k\rangle_{\mathrm{noiseless}}|/p$, where the noisy expectation value is computed for $p=10^{-2}$, and the Fermi momentum is $k_F=\pi/2$ (half-filling condition). As predicted by the theory, momenta close to the Fermi surface exhibit remarkable sensitivity to noise, while momenta further from the Fermi surface are stable to noise.
• Figure 2: Response to noise of the ground state of a $2D$ tight-binding model with Fermi surface. We represent the sensitivity to noise of the noisy expectation value $\langle n(k_x,k_y)\rangle_{\mathrm{noisy}}$ for a lattice of size $N=30 \times 30$, and occupation number $N_{\mathrm{occ}}=300$ (left, less than half-filling), $N_{\mathrm{occ}}=450$ (center, half-filling), $N_{\mathrm{occ}}=700$ (right, more than half-filling). The sensitivity to noise for a fixed momentum mode $(k_x,k_y)$ is estimated as the slope $| \langle n(k_x,k_y)\rangle_{\mathrm{noisy}}-\langle n(k_x,k_y)\rangle_{\mathrm{noiseless}}|/p$, where the noisy expectation value is computed for $p=10^{-2}$. As predicted by the theory, the momenta close to the Fermi surface exhibit remarkable sensitivity to noise, while the momenta further from the Fermi surface are stable to noise.
• Figure 3: Sketch of the circuits considered in Eq. \ref{['eq:noisy_time_evolution']}. Each circuit contains $d$ layers of local unitaries, and each layer of unitaries is followed by a layer of single-qubit noise (depicted by gray circle). For simplicity, we have depicted a circuit on $6$ qubits, where the unitaries are $3-$qubit gates. The unitary layer at time step $k$ is denoted by $\mathcal{U}_k$, and the noisy layer by $\Phi_k^{\mathrm{noisy}}=\mathcal{N}_p \circ \mathcal{U}_k$.

Theorems & Definitions (26)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 1
• Proposition 3
• proof
• Proposition 4
• proof