Near-Term Fermionic Simulation with Subspace Noise Tailored Quantum Error Mitigation

Abstract

Quantum error mitigation (QEM) has emerged as a powerful tool for the extraction of useful quantum information from quantum devices. Here, we introduce the Subspace Noise Tailoring (SNT) algorithm, which efficiently combines the cheap cost of Symmetry Verification (SV) and low bias of Probabilistic Error Cancellation (PEC) QEM techniques. We study the performance of our method by simulating the Trotterized time evolution of the spin-1/2 Fermi-Hubbard model (FHM) using a variety of local fermion-to-qubit encodings, which define a computational subspace through a set of stabilizers, the measurement of which can be used to post-select noisy quantum data. We study different combinations of QEM and encodings and uncover a rich state diagram of optimal combinations, depending on the hardware performance, system size and available shot budget. We then demonstrate how SNT extends the reach of current noisy quantum computers in terms of the number of fermionic lattice sites and the number of Trotter steps, and quantify the required hardware performance beyond which a noisy device may outperform classical computational methods.

Figures (8)

• Figure 1: Classical and quantum limits of the simulability of the 2D FHM. Left: The maximal number of Trotter steps achievable for a given QEM method at a fixed TQG fidelity, and a fixed $5\%$ root-mean-squared error (RMSE) of the site occupations. For more details see "Methods". Right: The required TQG fidelity for the simulation of a given FHM with SNT. The hatched region represents the approximate reach of classical computations as discussed in SI.
• Figure 2: Example of a decomposition of a parameterized multi-qubit Pauli operator $e^{i\theta \mathsf{XZY}}$ into a native gate set, consisting of arbitrary single-qubit rotations and a CZ as the native entangling gate. The red/orange blocks represent the noise channel $\mathcal{E}_k$ associated with implementing the Clifford unitary $\mathsf{U}_k^\mathrm{C}$. The layer of non-Clifford gates is comprised of a single-qubit rotation with angle $\theta_k$.
• Figure 3: Shot-by-shot representation of three different QEM methods. The (red)orange lightning bolts represent stochastically appearing (un)detectable errors. Gates with (red)orange crosses represent the gates added to the circuit in order to probabilistically cancel the (un)detectable errors using PEC. The parity check layer is highlighted in blue. If a detectable error propagates through the circuits and is subsequently detected via the ancilla measurement, the shot is discarded (indicated by an orange flag). Compared to PS, SNT circuits also utilize extra operations used to cancel undetectable errors, however only undetectable errors are canceled probabilistically while the detectable errors are removed by PS. The qualitative performance of each QEM method in terms of bias and cost is illustrated on the right.
• Figure 4: Squared bias (averaged over the site occupations) of the time evolution of a FHM with two sites after 10 Trotter steps as a function of CSP and the circuit error rate $\lambda$, for four different encodings and different mitigation schemes: no mitigation (blue), PS on local stabilizers (yellow), full SV including PP based on global stabilizers (green) and SNT (red). The dashed line represents a fit to the SNT data, assuming a second-order-error dominated bias $\propto \lambda^2$. The error bars represent a 1-$\sigma$ uncertainty due to a finite number of shots/circuits, which starts to dominate in the gray shaded area. The noisiness of the circuits is varied by changing the CZ gate fidelity. The insets display the simulated systems, with each node representing a qubit and different shades corresponding to different stabilizers.
• Figure 5: Optimal combinations of encoding and QEM, for the time evolution ($10$ Trotter steps) of a 2D FHM. The three black contour lines represent a CSP of 5%, 50% and 90% (left to right) of the circuit generated by the encoding with the smallest number of TQGs, depending on the system size. The white region in the top left corresponds to low circuit success probabilities, where a single parity check is not sufficient for a significant bias reduction, and the cost of PEC $C_\mathrm{PEC}^2 \gtrsim 10^6$ is too large for the given shot budget. Insets show the RMSE for optimal combinations of QEM and encoding.