Linked-Cluster Technique for Finding the Distance of a Quantum LDPC Code Alexey A. Kovalev Department of Physics & Astronomy University of California Riverside, CA 92521, USA Email: alexey.kovalev@ucr.edu Ilya Dumer Department of Electrical Engineering University of California Riverside, CA 92521, USA Email: dumer@ee.ucr.edu Leonid P. Pryadko Department of Physics & Astronomy University of California Riverside, CA 92521, USA Email: leonid@ucr.edu Abstract—We present a linked-cluster technique for calculating the distance of a quantum LDPC code. It offers an advantage over existing deterministic techniques for codes with small relative distances (which includes all known families of quantum LDPC codes), and over the probabilistic technique for codes with sufficiently high rates. I. I NTRODUCTION A practical implementation of a quantum computer will rely on quantum error correction (QEC) [1]–[3] due to the fragility of quantum states. There is a strong belief that surface (toric) codes [4], [5] can offer the fastest route to scalable quantum computation due to the error threshold around 1% and the locality of required gates [6]–[9]. Unfortunately, in the nearest future, the surface codes (in fact, any two-dimensional codes with local stabilizer generators [10]) can only lead to proof of the principle realizations as they encode a limited number of qubits (k), making any implementation of a useable quantum computer large (e.g., 2.2 × 10 8 physical qubits are required for a useful realization of Shor’s algorithm [11]). Lifting the restriction of locality but preserving the condi- tion that the stabilizer generators should only involve a limited number of qubits, one gets the quantum LDPC codes, or, more precisely, quantum sparse-graph codes [12], [13]. Unlike the surface codes, these more general quantum LDPC codes can have a finite rate. On the other hand, while there are no known upper bounds on the parameters of such codes, in practice, all families of quantum LDPC codes where the upper limit on the distance is known, have the distance scaling as a square root of the block length [14]–[17]. Nevertheless, such codes (in fact, any family of quantum or classical LDPC codes with limited weights of the columns and rows of the parity check matrix, and distance scaling as a power or a logarithm of the block length n) have a finite error probability threshold, both in the standard setting where syndrome is measured exactly, and with the syndrome measurement errors [18]. Given that non-local two-qubit gates are relatively inexpen- sive with floating gates [19], superconducting and trapped-ion qubits, as well as more exotic schemes with teleportation [20]– [25], a quantum computer relying on quantum LDPC codes is quite feasible. An example of a universal set of gates based on dynamical decoupling pulses for an arbitrary number of qubits with Ising couplings forming a bipartite graph (e.g., the Tanner graph corresponding to a quantum LDPC code) has been recently suggested by one of us [26]. Compared to general quantum codes, with a quantum LDPC code, each quantum measurement involves fewer qubits, mea- surements can be done in parallel, and also the classical processing could potentially be enormously simplified (note, however, that belief-propagation and related decoding algo- rithms that work so well for classical LDPC codes [27], [28] may falter in the quantum case [29]). Compared to surface codes, more general LDPC codes have higher rates, which translates in a large reduction of the total number of qubits necessary to build a useful quantum computer. Note that while our analytical threshold estimate in Ref. [18] is quite low, there are examples of quantum LDPC codes demonstrated to beat the bounded distance decoding limit [30]. Overall, it is quite plausible that the operation of quantum computers of the future will rely on (non-local) quantum LDPC codes. The very general proof [18] of the existence of a finite error probability threshold for quantum and classical LDPC codes with asymptotically zero relative distance is based on a simple observation that errors for such codes are likely to form small clusters affecting disjoint sets of stabilizer generators (parity check matrix rows). While the total weight of an error could be huge, the error can be surely detected if the size of each cluster is smaller than the code distance. Thus, in the case of the error detection, the threshold problem is related to the cluster size distribution for site percolation on a graph related to the Tanner graph of the code. In this work, we apply the idea of error clustering with LDPC codes to design a numerical algorithm for finding a distance of such a code. The basic principle is formulated in Theorem 1: to find the distance of a code, one only needs to check error configurations corresponding to connected error clusters. For any error weight w ≪ n, the number of such clusters is exponentially smaller than that of generic errors of the same weight. We consider the complexity of several well-known classical algorithms for finding code distance in application to quantum error correcting code. We conclude that the clustering algorithm beats deterministic techniques at sufficiently small relative distances (asymptotically at large n, all known families of quantum LDPC codes have zero relative distance), and the probabilistic technique for high-rate codes arXiv:1302.1845v1 [quant-ph] 7 Feb 2013