Fault-Tolerant Quantum Computation with Constant Error Rate

Abstract

Abstract. Shor has showed how to perform fault tolerant quantum computation when the probability for an error in a qubit or a gate, η, decays with the size of the computation polylogarithmically, an assumption which is physically unreasonable. This paper improves this result and shows that quantum computation can be made robust against errors and inaccuracies, when the error rate, η, is smaller than a constant threshold, ηc. The cost is polylogarithmic in space and time. The result holds for a very general noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with p > 2 states, namely qupits, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation.We use Calderbank-Shor Steane (CSS) quantum error correcting codes, generalized to qupits, and introduce a new class of CSS codes over Fp, called polynomial codes. It is shown how to apply a universal set of gates fault tolerantly on states encoded by general CSS codes, based on modifications of Shor's procedures, and on states encoded by polynomial codes, where the procedures for polynomial codes have a simple and systematic structure based on the algebraic properties of the code.Geometrical and group theoretical arguments are used to prove the universality of the two sets of gates. Our key theorem asserts that applying computation on encoded states recursively achieves fault tolerance against constant error rate. The generalization to general noise models is done using the framework of quantum circuits with density matrices. Finally, we calculate the threshold to be ηc ≃ 10 −6 , in the best case.The paper contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.1. Outline. Quantum computation has recently gained a lot of attention, due to oracle results [65,10,11] and quantum algorithms [25,65,36], in particular Shor's factorization algorithm [62]. These results indicate a possibility that quantum computers are exponentially more powerful than classical computers. It is yet unclear whether and how quantum computers will be physically realizable [19,22,30,37,50,51,55,59,69] but as any physical system, they in principle will be subjected to noise, such as decoherence [76,71], and inaccuracies. Thus, the question of correcting noise cannot be sepa...

Quantum error-correction failure distributions: Comparison of coherent and stochastic error models

“…We numerically simulate two common QEC d = 3 codes in this paper: the Steane [ [7,1,3]] code [2,10], and the titled-17 surface code [27][28][29]. Insertions of errors in the stochastic Pauli (SP) error model are treated as unitary qubit gates, so for both the SP error model and pulse-area error model model the evolution is purely unitary in our simulator.…”

“…The prescription here is for the Steane [ [7,1,3]] code with a few special considerations required for the surface code noted in Sec. III B.…”

“…The theory of fault tolerance and the associated threshold theorem show that given an error rate on physical qubits below some threshold, one can perform a quantum computation with arbitrary accuracy with manageable overhead due to error correction (see e.g., [1][2][3][4][5][6][7][8][9]). It is well known that the Pauli matrices form a basis for any arbitrary qubit state.…”

“…To show this, we compare the output failure distributions of a quantum error correction (QEC) memory circuit correct-ing stochastic Pauli errors to coherent errors. We utilize a numerical simulation of the entire encoded wavefunction, and we examine the failure distribution of the logical state for the Steane [ [7,1,3]] code and a distance 3 surface code. We devise failure metrics that avoid the need to implement a final round of perfect error correction and decoding, commonly used when studying the failure characteristics of a logical channel [18,22,23,26].…”

Contagious error sources would need time travel to prevent quantum computation

“…If we accept quantum mechanics (more precisely, quantum probability), then at the theoretical level this question is usually interpreted as the fault tolerance problem: Can a quantum computer still work if all of its gates and qubits are noisy? There are by now various fault-tolerance theorems for quantum computation, which establish that reliable quantum computation is indeed possible in principle assuming that the noise present in different qubits or gates is quasi-independent [1,8,11], and is below some threshold error rate. This threshold is called the fault tolerance constant.…”