Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent X and Z noise.
Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent X and Z noise.
This thesis deals with the study of computational complexity of decoding stabilizer codes. The first three chapters contain all the necessary background to understand the main result of this thesis. First, we explain the necessary notions in computational complexity, introducing P, NP, #P classes of problems, along with some examples intended for physicists. Then, we explain the decoding problem in classical error correction, for linear codes on the binary symmetric channel and discuss the celebrated result of Mcleicee et al., in [1]. In the third chapter, we study the problem of quantum communication, over Pauli channels. Here, using the stabilizer formalism, we discuss the concept of degenerate errors. The decoding problem for stabilizer codes, which simply neglects the presence of degenerate errors, is called quantum maximum likelihood decoding (QMLD) and it was shown to be NP-complete, by Min Hseiu Heish et al., in [2]. We focus on the problem of optimal decoding, called degenerate quantum maximum likelihood decoding (DQMLD), which accounts for the presence of degenerate errors. We will highlight some instances of stabilizer codes, where the presence of degenerate errors causes drastic variations between the performances of DQMLD and QMLD. The main contribution of this thesis is to demonstrate that the optimal decoding problem for stabilizer codes is much harder than what the previous results had anticipated. In the last chapter, we present our own result (in Thm. 5.1.1), establishing that the optimal decoding problem for stabilizer codes, is #P-complete. To prove this, we demonstrate that the problem of evaluating the weight enumerator of a binary linear code, which is #P-complete, can be reduced (in polynomial time) to the DQMLD problem, see (Sec. 5.1). Our principal result is also presented as an article in [3], which is currently under review for publication in IEEE Transactions on Information Theory. In addition to the main result, we also show that under certain conditions, the outputs of DQMLD and QMLD always agree. We consider the conditions developed by us to be an improvement over the ones in [4, 5].
Quantum error correction is instrumental in protecting quantum systems from noise in quantum computing and communication settings. Pauli channels can be efficiently simulated and threshold values for Pauli error rates under a variety of error-correcting codes have been obtained. However, realistic quantum systems can undergo noise processes that differ significantly from Pauli noise. In this paper, we present an efficient hard decoding algorithm for optimizing thresholds and lowering failure rates of an error-correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. We use our hard decoding algorithm to study the performance of several error-correcting codes under various non-Pauli noise models by computing threshold values and failure rates for these codes. We compare the performance of our hard decoding algorithm to decoders optimized for depolarizing noise and show improvements in thresholds and reductions in failure rates by several orders of magnitude. Our hard decoding algorithm can also be adapted to take advantage of a code's non-Pauli transversal gates to further suppress noise. For example, we show that using the transversal gates of the 5-qubit code allows arbitrary rotations around certain axes to be perfectly corrected. Furthermore, we show that Pauli twirling can increase or decrease the threshold depending upon the code properties. Lastly, we show that even if the physical noise model differs slightly from the hypothesized noise model used to determine an optimized decoder, failure rates can still be reduced by applying our hard decoding algorithm.
The decoding problem is a ubiquitous algorithmic task in fault-tolerant quantum computing, and solving it efficiently is essential for scalable quantum computing. Here, we prove that minimum-weight decoding is NP-hard in three quintessential settings: (i) the color code with Pauli Z errors, (ii) the surface code with Pauli X, Y and Z errors, and (iii) the surface code with a transversal CNOT gate, Pauli Z and measurement bit-flip errors. Our results show that computational intractability already arises in basic and practically relevant decoding problems central to both quantum memories and logical circuit implementations, highlighting a sharp computational complexity separation between minimum-weight decoding and its approximate realizations.
Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the X- and Z-syndromes. For color codes under independent X- and Z-error models, we consider Separate Minimum-Weight decoding.
Assuming P=NP, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the 4.8.8 color code on the torus, there exists a constant c>0 such that no polynomial-time algorithm can always produce a solution whose weight is within cN1/14 of the optimum, where N is the number of qubits, unless P=NP. For the planar surface code, we obtain an Ω(N1/18) gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT.
Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.
We study the exact maximum-likelihood decoding of quantum error-correcting codes. Our primary focus is on the repetition code under circuit-level noise, which serves as a fundamental basis for quantum error correction experiments and is the only code that has achieved large distances and extremely low error rates. We uncover that the repetition code under circuit-level noise has an exact solution and propose an optimal maximum-likelihood decoding algorithm called "planar." This algorithm is based on the exact solution of the spin-glass partition function on planar graphs and has polynomial computational complexity. Through extensive numerical experiments, we demonstrate that our algorithm uncovers the exact threshold for depolarizing noise and realistic superconductor SI1000 noise. Furthermore, we apply our method to analyze data from recent quantum memory experiments conducted by Google Quantum AI, revealing that part of the error floor was attributed to the decoding algorithm used by Google. Finally, we implemented the repetition code quantum memory on superconducting systems using a 72-qubit quantum chip lacking reset gates, demonstrating that even with an unknown error model, the proposed algorithm achieves a significantly lower logical error rate than the matching-based algorithm. We also discuss the application of our approach to quantum codes whose maximum-likelihood decoding problem corresponds to a spin-glass problem, including the surface code under independent code-capacity noise.
We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time O(n2), where n is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time O(nχ3) where χ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension χ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for χ≥4.