Quantum Error Mitigation Advances NISQ Quantum Computing
Error Mitigation Needed in the NISQ Era: Quantum Computing's Limits As quantum hardware develops rapidly, researchers are using Quantum Error Mitigation (QEM) software to solve quantum computers' noise, imperfect interactions, and elementary physical component errors. Quantum Error Correction (QEC) is the long-term solution, but full-scale fault tolerance may require millions of qubits for industrial applications. QEM bridges Noisy, Intermediate-Scale Quantum (NISQ) devices to enable immediate quantum information processing advances.
Quantum Error Mitigation QEM approaches address noise without hardware expansion or a strict error threshold, unlike QEC. QEM uses post-processing on ensemble circuit run outputs to reduce noise bias in observable expectation values. Common NISQ applications like variational quantum circuits or approximate optimisation approaches use short-depth circuits to estimate expectation values. Bias and Sampling Overhead: The Main Issue A noisy quantum computer's results are limited by the Mean Square Error (MSE), which comprises statistical error (variance) and systematic error (bias). The bias is a systematic shift that persists with infinite sampling, but shot noise (variance) can be reduced by increasing circuit executions. Quantum Error Mitigation QEM approaches aim to reduce this bias by building an estimator. However, minimising bias frequently increases estimator variance, making this benefit costly. This trade-off is shown by the sampling overhead, or the number of circuit runs needed to match the untreated noisy estimator's shot noise. A important discovery reveals QEM's inherent limitations: sample overhead grows exponentially with the average number of failures per circuit run, or circuit fault rate. Due to exponential scaling, QEM alone is unlikely to work for circuits with high failure rates.
Different Error-Mitigation Methods QEM has numerous techniques, each tailored to a certain noise and ideal state assumption: ZNE, also known as error extrapolation, obtains noisy expectation values at various boosted error rates and projects them back to the imaginary zero-noise limit. A simple polynomial function like Richardson extrapolation can approach expectation value degradation when the circuit defect rate is low. Identity-equivalent gates and pulse stretching increase noise. The prominent Quantum Error Mitigation QEM technology ZNE allows cutting-edge simulations with up to 127 qubits. Probabilistic Error Cancellation (PEC): PEC is special because it eliminates expectation value bias. PEC defines the optimal (noise-free) quantum channel as a linear combination of hardware-implemented physical, noisy basis operations. Monte Carlo sampling estimates quasi-probability decomposition. If Pauli noise impacts every gate, sampling overhead increases exponentially, which is costly. PEC requires substantial noise channel knowledge, which is commonly obtained by tomography. Measurement Error Mitigation (MEM): MEM covers SPAM mistakes during state preparation and final measurement and is typical in near-term research. Assuming measurement noise is primarily inside the computational domain, the assignment matrix, which describes transition probabilities between ideal and measured outcomes, can be inverted to correct the ideal output distribution. Symmetry Constraints (SYM): This method uses natural symmetries like particle number and parity. Either direct post-selection (removing runs that don't pass a symmetry check) or post-processing to obtain a symmetry-verified expectation value can suppress mistakes that violate these symmetries. This method is cost-effective since sample overhead decreases with success rate. PUR/VD: For algorithms that want a pure end state. The expected value of a purified state is estimated using VD (Error Suppression by Derangement, or ESD), which converges exponentially rapidly to the noisy state's dominating eigenvector as the number of copies increases. This purification is done by measuring the cyclic permutation operator across noisy state copies. Beyond Physical Errors Quantum Error Mitigation QEM is beneficial beyond hardware noise reduction. It is also effective for resolving algorithmic (compilation) errors due to the algorithm itself. Unlike physical faults, these errors cannot be erased by QEC and occur even with perfect hardware. ZNE can be used to extrapolate the infinite-order (zero algorithmic error) Hamiltonian simulation result via Trotterization over numerous levels. QEM is also important for quantum computing's future, improving QEC in the early fault-tolerant stage. In a fault-tolerant system, residual logical errors (errors that evade QEC) can be reduced by using Quantum Error Mitigation QEM techniques like PEC and the Pauli frame update to cancel logical Pauli gates. This reduces physical qubit overhead. In conclusion To optimise existing and future quantum technology, QEM methods are important. Quantum Error Mitigation QEM offers customised ways from algebraically inverting noise channels (PEC) to extrapolating experimental data (ZNE) and using structural features (SYM, PUR) to extract high-fidelity results despite physical restrictions. Research is ongoing to systematise and refine these strategies to find the optimum hybrid QEM method for quantum advantage.
