Quantum error correction in the presence of biased noise

Have you ever heard about error correction? Without it, we could not obtain awe-inspiring pictures of Jupiter and its moons, conduct intelligible mobile phone calls, or have reliable computers. In this blog post, we explain the basic ideas behind error correction and how to apply it to quantum computing. In addition, we discuss how we can use so-called biased noise in quantum computers to our advantage in order to improve the performance of quantum error-correcting protocols.

Basic ideas behind error correction
Error correction techniques strive to protect information from the detrimental effects of noise that may change or even completely destroy it. Let us consider a concrete example of storing one bit of information, either 0 or 1, in the presence of noise. One way in which noise can corrupt the stored information is through a bit-flip error, i.e., the value of the bit is changed from 0 to 1 or from 1 to 0. We assume that bit-flip errors are not too likely, e.g., each bit flip happens independently with probability p=0.02, and that we do not know when they occur. Then, how can we store one bit?

There is a simple error correction strategy that relies on using more resources and making multiple copies of the bit that we want to protect against bit-flip errors. Instead of keeping only one copy of the bit, either 0 or 1, we choose to store a bit string with, for instance, five copies of that bit, either 00000 or 11111, respectively. Then, if we saw a bit string 01000, we could take the majority vote of the bit values and guess that the second bit might have suffered from the bit-flip error. Subsequently, we could reliably infer the value of the stored bit to be 0. It is easy to see that we would succeed as long as fewer than a half of the bits in the bit string suffered from the bit-flip errors and changed their values. A similar strategy would work if bits also suffered from erasure errors – when an erasure error happens, then the corresponding bit is irrevocably lost and we know it. We can then represent the value of this bit by a symbol *. For instance, as long as not all the bits are erased, we still preserve the value of the bit that we want to protect. In general, erasure errors are easier to correct than bit-flip errors because we know which bits have been lost. Also, we can consider a scenario where both erasure and bit-flip errors happen. For instance, if we saw a bit string 10*00, then we would know that the third bit had been erased and we would also suspect that the first bit had suffered from the bit-flit error. This, in turn, would allow us to infer the value of the stored bit to be 0.

Can we protect quantum bits?
Perplexingly, the strategy that we just discussed is inadequate to protect quantum bits (or qubits for short). This is due to quantum noise being more complex than classical noise and the no-cloning theorem [1], which asserts that it is impossible to make a copy of a qubit in an arbitrary unknown state. Unlike a classical bit that is either 0 or 1, a qubit can be in any state α|0〉 + β|1〉, which is a superposition of two states |0〉 and |1〉 (where α and β are two complex numbers, such that the squares of their modules sum up to 1). A qubit may suffer not only from bit-flip noise, but also from phase-flip noise. Similarly to its classical counterpart, a bit-flip error changes the qubit state from α|0〉 + β|1〉 to β|0〉 + α|1〉. A phase-flip error, however, has no classical counterpart and it changes the qubit state α|0〉 + β|1〉 to α|0〉 – β|1〉. In addition to quantum noise being more complex than classical noise, we cannot learn the state of the qubit without drastically altering it. We assume that bit-flip and phase-flip errors are not too likely to happen, e.g., each bit-flip and phase-flip error happens independently with probability p=0.02, and that we do not know when they occur. We may then wonder whether, despite these challenges, it is possible to protect qubits from bit-flip and phase-flip errors simultaneously.

Figure 1. The surface code can be realized by placing qubits on the vertices of the square lattice and measuring Pauli X and Z parity checks associated with dark and light faces, respectively.

Quantum error correction
In the mid-nineties, Peter Shor and Andrew Steane gave an affirmative answer to this question [2,3]. In particular, they provided the very first quantum error correction (QEC) strategies. With their seminal work, the field of QEC was born and many QEC strategies have been developed since then. One of the best performing QEC strategies is based on topological QEC codes, such as the surface code [4]; see Figure 1. As we will see later in the blog post, the surface code plays a central role in our research efforts at the AWS Center for Quantum Computing.

Typically, QEC protocols are designed and optimized assuming that the probability of the bit-flip and phase-flip error is comparable. Such an assumption is very natural from the theoretical perspective, since it is general and it does not require any detailed knowledge of the physical origins of noise. On the other hand, one may be able to improve the performance of QEC protocols (measured in terms of logical error rate) by leveraging the noise structure, if there is any.

Exploiting noise bias
In a series of articles [5,6,7] last year, we explored QEC protocols based on the surface code and its various generalizations. We were interested in the scenarios where noise exhibits imbalance (or “bias”) between bit-flip and phase-flip errors. Such noise bias can be realized with bosonic qubits, for example. Importantly, our optimizations invoked only minor adjustments to the standard protocols, such as Clifford deformations of the parity checks that we measure. We emphasize that Clifford deformations are broadly compatible with and easily adaptable to the usual hardware implementations of QEC protocols. We found that the performance of QEC protocols, measured in terms of the logical error rate, can be significantly improved if we use the knowledge and the structure of the noise bias. For instance, for noise with error rate p=0.02 and bias of η=1000 (i.e., where the phase-flip errors are 1000x more likely to happen than bit-flip errors), the logical error rates for the best and worst performing Clifford-deformed surface codes of distance three differ by two orders of magnitude. Subsequently, to suppress logical errors to the same level, fewer resources are needed for optimized Clifford-deformed surface codes compared to the off-the-shelf surface code.

So far, we have discussed bit-flip and phase-flip noise, which is a convenient model to analyze and simulate the performance of QEC. However, in reality, quantum noise may have different structure that we can also exploit. In many quantum technologies, such as superconducting circuits, the states |0〉 and |1〉 of the qubit are chosen as the ground state and the first excited state of the physical system, such as a transmon, that we use to define a qubit. Subsequently, the dominant type of noise is the amplitude damping noise that captures energy relaxation from the excited state to the ground state. When the amplitude damping error happens, it transforms the state |1〉 into the state |0〉, without affecting the state |0〉. It is thus natural to ask whether using the knowledge of such noise bias we can simplify QEC protocols and reduce the resources required to implement them even further.

In our recent work [8], we established a new paradigm of exploiting the bias between the amplitude damping and other types of quantum noise. We provided a simple QEC strategy to detect amplitude damping errors and convert them into erasure errors, boosting the performance of fault-tolerant protocols. We proposed two realizations of our scheme with superconducting circuits, and discussed in detail the implementation of gates and dephasing mechanisms.

Figure 2. Comparison of the correctable region for the surface code with the standard scheme (dark area) and our scheme (dark/light area). Our scheme tolerates more noise (described by two parameters e and p) than the standard scheme.

With our scheme, we can overcome the conventional limit on fidelity due to amplitude damping noise and, subsequently, enhance the performance of fault-tolerant protocols. Our scheme is also well-suited to deal with leakage errors as they can be detected and converted to erasures [9]. We demonstrate the usefulness of our scheme by simulating the memory threshold of the surface code under the circuit noise. We observe that, e.g., the correctable region where our scheme works is approximately 3.5x larger than the correctable region for the standard protocol; see Figure 2. Since our scheme efficiently suppresses amplitude damping noise, thus in order to improve the performance of fault-tolerant protocols further, the engineering efforts should focus on extending the dephasing time and the quality of quantum coherent control.

To summarize, biased noise can be very beneficial from the perspective of quantum error correction, leading to the reduced resource requirements. We believe that there are many more clever ways to leverage the knowledge of the noise affecting quantum devices, and our works only scratch the surface. If you find this blog post interesting and would like to learn more, then we encourage you to have a look at our articles that are linked in bibliography.

Bibliography
1. W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802–803 (1982)
2. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A 52, R2493(R) (1995)
3. A. Steane, Error Correcting Codes in Quantum Theory, Physical Review Letters 77, 793 (1996)
4. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory Journal of Mathematical Physics 43, 4452-4505 (2002).
5. A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, Clifford-deformed Surface Codes, arXiv.2201.07802 (2022).
6. O. Higgott, T. C. Bohdanowicz, A. Kubica, S. T. Flammia, and E. T. Campbell, Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise, arXiv:2203.04948 (2022).
7. Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, Tailored XZZX codes for biased noise, arXiv:2203.16486 (2022).
8. A. Kubica, A. Haim, Y. Vaknin, F. Brandão, and A. Retzker, Erasure qubits: Overcoming the T_1 limit in superconducting circuits, arXiv.2208.05461 (2022).
9. Y. Wu, S. Kolkowitz, S. Puri, and J. D. Thompson, Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays, Nature Communications 13, 4657 (2022)