Motivation for quantum error correction

Motivation for quantum error correction#

This textbook gives a hands-on introduction to concepts in quantum error correction (QEC). But why bother about QEC in the first place, and why hands-on?

The answer is quite simple.

Resource estimates of quantum algorithms that are expected to provide significant advantage over their classical counterparts – factoring at cryptographic scales, quantum chemistry for catalysis and materials, and physics simulations – all point to billions to tens of billions of logical operations for a successful computation. That implies logical errors per operation on the order of 1 part in \(10^{10}\) or better – without this, we will never be able to run billions of operations successfully. For example, RSA-2048 factoring in the abstract circuit model requires roughly \(3\times10^9\) operations before error-correction overheads. Modern quantum chemistry (e.g., qubitization and tensor hypercontraction) compiles FeMoco simulations to billions of logical operations running for several days on multi-million-qubit surface-code machines (see refs. [7–8]).

By contrast, the best quantum systems today typically achieve best-in-class two-qubit error rates (which are typically the limiting errors) between 1 part in \(10^4\) and 1 part in \(10^3\), depending on the qubit architecture and operating conditions (see refs. [1-6]). These error rates are typically referred to as “three nines” (\(99.9\%\)) to mean an error of \(0.1\%\) or “four nines” (\(99.99\%\)) to mean an error of 0.01%.

QEC is the bridge between these regimes – taking us from error rates of order 1 in 10,000 to error rates of order 1 in 10 billion. This is a gap with 6 orders of magnitude.

QEC is able to fill this gap because it converts incremental hardware improvements into exponential reductions in error. Later in this textbook, you will learn the concept of code distance. For now, it suffices to think of it as the level of repetition or redundancy needed for error correction. For distance-\(d\) codes, a useful heuristic is

\[ p_L \sim (c p)^{(d+1)/2} \]

where \(p_L\) is the logical probability of error with QEC enabled, \(p\) is the probability of error without error correction, and \(c\) is a constant due to the overhead of QEC. This scaling means reducing \(p\) by a factor of two can yield orders of magnitude of logical suppression at larger code distances \(d\). In this way, QEC is the lever that turns today’s \(\sim10^{-3}-10^{-4}\) two-qubit errors into the \(10^{-10}\) logical error-per-operation needed for long, valuable quantum programs. In the literature, the first destination on the way to billions of logical operations is the so-called MegaQuOp (millions of quantum operations) regime, with logical error rates at the part-per-million level (see ref. [9]).

Implementing quantum error correction in practice#

There is a significant difference between understanding QEC and doing QEC. Some of the key concepts in QEC – stabilizers, syndromes, and decoders – can be understood entirely with pencil-and-paper algebra. But implementing QEC on devices requires understanding the quantum circuits in detail: where measure qubits live, how checks are pipelined, latency of feed-forward, throughput of readout, control-system timing budgets, and architectural constraints (coupling maps, crosstalk, calibration cadence). This textbook therefore introduces QEC concepts hands-on at the circuit level and aims to discuss these details to encourage QEC experimentation and discovery of novel error-correcting codes. The goal is to demonstrate both why QEC works and how to run it.

References#

[1] Kjaergaard et al. (Annu. Rev. Condens. Matter Phys., 2020): Superconducting Qubits: Current State of Play. DOI: 10.1146/annurev-conmatphys-031119-050605. arXiv:1905.13641.
[2] Ballance et al. (PRL, 2016): High-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits. DOI: 10.1103/PhysRevLett.117.060504. arXiv:1512.04600.
[3] Sung et al. (PRX, 2021): Realization of High-Fidelity CZ and ZZ-Free iSWAP Gates with a Tunable Coupler. DOI: 10.1103/PhysRevX.11.021058. arXiv:2011.01261.
[4] Huang et al. (Nature, 2019): Fidelity benchmarks for two-qubit gates in silicon. DOI: 10.1038/s41586-019-1197-0. arxiv:1805.05027.
[5] Evered et al. (Nature, 2023): High-fidelity parallel entangling gates on a neutral-atom quantum computer. arXiv:2304.05420.
[6] Quantinuum (Blog, 2024): Quantinuum extends its significant lead in quantum computing, achieving historic milestones for hardware fidelity and Quantum Volume Blog.
[7] Gidney & Ekera (Quantum, 2021): How to factor 2048-bit RSA integers in 8 hours using 20 million noisy qubits. Quantum 5, 433. arXiv:1905.09749
[8] Lee et al. (PRX Quantum, 2021): Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction. DOI: 10.1103/PRXQuantum.2.030305. arXiv:2011.03494.
[9] Preskill: Beyond NISQ: The Megaquop Machine. arxiv:2502.17368.

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