Volume Thresholds for Quantum Fault Tolerance Vaneet Aggarwal, 1 A. Robert Calderbank, 1 Gerald Gilbert, 2 and Yaakov S. Weinstein 2 1 Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA 2 Quantum Information Science Group, Mitre, 260 Industrial Way West, Eatontown, NJ 07224, USA We introduce finite-level concatenation threshold regions for quantum fault tolerance. These volume thresholds are regions in an error probability manifold that allow for the implemented system dynamics to satisfy a prescribed implementation inaccuracy bound at a given level of quantum error correction concatenation. Satisfying this condition constitutes our fundamental definition of fault tolerance. The prescribed bound provides a halting condition identifying the attainment of fault tolerance that allows for the determination of the optimum choice of quantum error correction code(s) and number of concatenation levels. Our method is constructed to apply to finite levels of concatenation, does not require that error proabilities consistently decrease from one concatenation level to the next, and allows for analysis, without approximations, of physical systems characterized by non-equiprobable distributions of qubit error probabilities. We demonstrate the utility of the new method via a general error model. PACS numbers: 03.67.Lx, 03.67.Pp The theory of quantum fault tolerance allows for suc- cessful quantum computation despite faults in the imple- mentation of the quantum computing machine [1, 2]. To achieve fault tolerance, quantum information may be en- coded into logical qubits, and correspondingly encoded quantum gates implemented between the logical qubits such that the quantum information is protected. Protec- tion against errors can be achieved via error correcting or error avoiding techniques [3–5]. Such techniques may be concatenated [6] to increase robustness of the quan- tum information. To ensure fault-tolerant operation, ba- sic gates must be implemented with an error probabil- ity below a critical threshold value the determination of which has been an important research focus over the past decade [7–10]. Standard fault tolerance requires that the error probability decreases exponentially as a function of concatenation level. The error threshold is then deter- mined by the maximum allowable initial error probability such that the error probability goes to zero in the limit of infinite concatenation [11]. In this paper we present a formulation of fault tolerance based on threshold re- gions defined in terms of finite-level concatenations, thus avoiding reliance on unrealistic infinite limits, and that does not require decrease of error probabilities at each concantenation. We find that this leads to a more accu- rate and powerful formulation of fault tolerance. Our approach to fault tolerance requires that a pre- scribed bound on the accuracy of a quantum computa- tional process be satisfied. This bound serves as a halting condition which helps determine the necessary number of quantum error correction (QEC) code concatenation levels, optimal QEC code(s) and other resource require- ments. Having a bound also allows us to forgo the re- quirement of an exponential reduction in error probabil- ity upon each level of concatenation. We present exam- ples where prescribing a halting condition leads to sav- ings in the number of qubits by properly choosing QEC codes and where, upon concatenation, the accuracy of the quantum computation decreases before eventually in- creasing and satisfying the halting condition. Given a quantum computational process we define the implementation inaccuracy as the norm difference be- tween the output density matrix of the ideally defined and actually implemented quantum evolutions with a supremum taken over all initial density matrices. We obtain a closed form expression for the implementation inaccuracy for the case of protecting one logical qubit of information from arbitrary, non-equiprobable σ x , σ y , and σ z errors (where σ i are the Pauli spin matrices). Based on the above our approach to fault tolerance is defined by: (1) utilizing a prescribed operator-theoretic halting condition that allows us to properly calculate the optimal number of concatenation levels and determine optimal error correction codes, (2) describing as accu- rately as possible the exact error dynamics of both the system and the necessary QEC codes. We thus consider the evolution of the entire error manifold allowing all (including possible non-equiprobable) errors, and (3) be- cause the sole fault tolerance criterion is to fulfill the halting condition we do not require that error probabil- ities decrease at each concatenation level. The baseline approach [10] is a special case of our more general and accurate approach. QEC is a method of detecting and correcting the errors that degrade quantum information [3]. The general error correction condition of a QEC code is: M {c→l} (V dec RεV enc · (M {l→c} (ρ))) = ρ, (1) where ρ is the initial density matrix defined on the space of logical qubits. The linking maps, M {c→l} and M {l→c} mathematically connect the logical and computational Hilbert spaces but do not represent physical processes. The map V enc on the computational Hilbert space per- forms the actual encoding into the QEC code space. The dynamical map ε describes the errors that affect the en- coded state. R is the syndrome measurement and recov- ery operation, and V dec is the decoding operation cor- responding to V enc , which yields the final state in the