Error mitigation and circuit division for early fault-tolerant quantum phase estimation

TL;DR

The paper tackles the challenge of enabling useful quantum computations in the near term by formulating an end-to-end framework for early fault-tolerant quantum computing that jointly optimizes error correction, error mitigation, and algorithm design. It introduces a robust QPE approach, MSQPE, based on circuit division of the QFT-based QPE circuit, and couples it with EUMLE to extend error mitigation to arbitrary noise channels, providing convergence guarantees and quantifiable overheads. The work delivers concrete results, including cost and resource estimates for Hubbard and molecular Hamiltonians, a detailed comparison with single-control RPE, and a comprehensive resource-trade-off analysis that reveals an elbow-like regime where modest qubit reductions yield meaningful runtime benefits. These contributions collectively offer an end-to-end view of how early fault-tolerance may reduce physical-qubit requirements at the cost of increased wall-clock time, and they identify concrete directions for improving the practicality of early FT quantum computation.

Abstract

As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of "early fault tolerance" where limited error correction is available. We propose a framework for designing early fault-tolerant algorithms by trading between error correction overhead and residual logical noise, and apply it to quantum phase estimation (QPE). We develop a quantum-Fourier-transform (QFT)-based QPE technique that is robust to global depolarising noise and outperforms the previous state of the art at low and moderate noise rates. We further introduce the Explicitly Unbiased Maximum Likelihood Estimation (EUMLE), a data processing technique that mitigates arbitrary errors in QFT-based QPE schemes. EUMLE provides consistent, asymptotically normal error-mitigated estimates, addressing the open problem of extending error mitigation beyond expectation value estimation. Applying this scheme to the ground state problem of the two-dimensional Hubbard model and various molecular Hamiltonians, we find we can roughly halve the number of physical qubits with a $\sim 10\times$ wall-clock time overhead, but further reduction causes a steep runtime increase. This work provides an end-to-end analysis of early fault-tolerance cost reductions and space-time trade-offs, and identifies which areas can be improved in the future.

Figures (10)

• Figure 1: Illustration of our framework for designing algorithms for early fault-tolerance and how it differs from traditional fault-tolerant algorithm design and resource estimation. The standard approach (left) involves optimizing an algorithm in the absence of noise and then choosing quantum error correction parameters such that the failure probability is small enough. We propose an alternative approach better suited to early fault-tolerance. By using noise resilient algorithms as a building block, we can allow for non-negligible levels of noise throughout the execution of a circuit, jointly optimizing the parameters of the algorithm, the quantum error correcting scheme, and the error mitigation technique used to address the residual error.
• Figure 2: Ratio of the total executions times $\mathcal{T}_{\mathrm{tot}}$ of the optimized robust phase estimation (RPE) algorithm of Ref. belliardoAchieving2020 and the maximum-likelihood sin-state quantum phase estimation (MSQPE) algorithm developed in this work across a range of target errors on the phase estimate $\epsilon$ and noise rates $\gamma$ translating to an error rate per unitary of $p_{\mathrm{err}}=(1-e^{-\gamma})$. The dashed line at $1$ denotes equivalent performance between the two methods; points above $1$ give a region where the MSQPE estimator performs better, points below $1$ show where RPE outperforms MSQPE.
• Figure 3: Physical costs for MSQPE applied to (left) the qubitized Fermi-Hubbard model Hamiltonian and (right) the active-space molecular Hamiltonians of some chemical systems. For the Fermi-Hubbard Hamiltonian, we choose the hopping parameter $t=1$ to set the energy units and interaction strength $u/t = 4$. For the electronic structure Hamiltonians, the active spaces sizes (number of electrons and spatial orbitals) are notated in the legend. The target precision on the energy is $\Delta E = 10^{-2}$ for the Hubbard model and $\Delta_E = 10^{-3} E_\text{h}$ for the chemical systems. The physical resources are estimated assuming computation in the surface code using CCZ resource states produced by a single factory; the size of surface code and CCZ factory parameters are chosen to minimize computation volume while keeping the error rate below a choosen $\gamma$, assuming physical error rate of $10^{-3}$, a surface code clock cycle of $1µs$, and $50\%$ routing overhead. The residual error is assumed to follow a global depolarizing noise model. Changing $\gamma$ allows to trade between the number of physical qubits (used as resource for implementing better error correction) and total runtime. For comparison, we report the physical costs of the standard fault-tolerant single-circuit implementation of sin-state QPE, accepting a failure probability of $1\%$ or $0.1\%$ (triangles).
• Figure 4: (top) Maximum circuit depth for a circuit with 100 logical qubits and an effective sampling rate of one sample per minute over a variety of physical error rates and qubit numbers. We assume a circuit composed entirely of CNOT gates, a raw two-qubit gate time of 20 nanoseconds, and a surface code cycle time of 970 nanoseconds. Errors are mitigated using probabilistic error cancellation (PEC), or by a combination of surface code quantum error correction and PEC. We maximize the achievable depth by optimizing over the choice of whether or not to use error correction, and the code distance. (bottom) Increase in circuit depth allowed by this combination of QEC and EM, when compared with a protocol that either uses EM or QEC (with QEC parameters chosen such that the error probability throughout the circuit is less than $1\%$). The existence of the dark band reveals a regime where this increase is substantial.
• Figure 5: Sin-state QPE circuit with control dimension $K$. First, the state $\ket{s_K}$ is prepared on the control register of $\lceil\log_2K \rceil$ qubits (conditional on $1$ measured the ancillary qubit) , and the system register is prepared in state $\ket{\phi}$. Then, controlled unitary $c_K U = \sum_{k=0}^{K-1} \ket{k}\bra{k} \otimes U^k$ is applied. Finally, $QFT^\dagger$ is applied on the control register, and the register is measured in the computational basis to obtain a bit string $x$. The oracular cost is $K-1$.

Theorems & Definitions (24)

• Theorem 1: Thm. \ref{['thm:mle_sinqpe']}, informal
• Theorem 2: Thm. \ref{['thm:em_overhead_qpe']}, informal
• Theorem 3: Thm. \ref{['thm:eumle']}, informal
• Definition A.4
• Definition B.5: QPE for single eigenstates
• Definition B.6: noiseless QPE for single eigenstates
• Theorem B.8
• Theorem B.11
• Definition C.12: QPE in the presence of depolarising noise
• Theorem C.15