Fault-tolerant quantum computation with constant error

Abstract

In the past year many developments have taken place in the area of quantum error corrections. Recently Shor showed how to perform fault tolerant quantum computation when,~, the probability for a fault in one time step per qubit or per gate, is polylogarithmically small. This paper closes the gap and shows how to perform fault tolerant quantum computation when the error probability, q, is smaller than some constant threshold, q.. The cost is polylogarithmic in time and space, and no measurements are used during the quantum computation. The same result is shown also for quantum circuits which operate on nearest neighbors only.To achieve this noise resistance, we use concatenated quantum error correcting codes. The scheme presented is general, and works with any quantum code, that satisfies certain restm"ctions, namely that it is a "proper quantum code". The constant threshold r10 is a function of the parameters of the specifc proper code used. We present two explicit classes of proper quantum codes. The first class generalizes classical secret sharing with polynomials.The codes are defined over a field with p elements, which means that the elementary quantum particle is not a qubit but a "qupit". The second class uses a known class of quantum codes and converts it to a proper code.We estimate the threshold qO to be = 10-6. Hopefully, this paper motivates a search for proper quantum codes with higher thresholds, at which point quantum computation becomes practical.

Quantum computing with realistically noisy devices

“…The simplest purification method known so far involves using 15 prepared |π/8 states. One is encoded into the [ [7,1,3]] code 44 (a code that encodes 1 qubit in 7 qubits with minimum distance 3, which implies that it can correct any (3 − 1)/2 = 1 qubit error or detect any (3 − 1) = 2 qubit errors). The other 2 × 7 |π/8 = 14 states are used to implement a conditional logical HAD from an ancilla to realize an encoded HAD measurement.…”

“…We conjecture that by using state injection with Steane's fault-tolerant methods for preparing states, the additional error on the purified |π/8 state is dominated by a decoding error of the order of the logical CNOT error. Specifically, one can encode one noisy logical |π/8 by teleportation into the [ [7,1,3]] code using a Bell state correlating a logical qubit and a [ [7,1,3]]-encoded qubit. (Strictly speaking, our architecture requires the use of logical qubit pairs associated with blocks of the C 4 /C 6 codes, but we treat each qubit in a pair identically.)…”

“…It can therefore be well verified using Steane's methods. The error in the state teleported into the [ [7,1,3]] code is due to the initially prepared |π/8 state, initial error in the logical qubit of the Bell state used, and the CNOT and measurements needed for the teleportation Bell measurement. Because we are operating with logical qubits of the C 4 /C 6 architecture, all but the first of these errors are comparatively small, assuming that the C 4 /C 6 encoding level is chosen so as to significantly decrease CNOT errors.…”

“…The latter event could cause unintentional acceptance of the final state, but only if additional error occurred elsewhere. At the end of the procedure, the [ [7,1,3]]-encoded qubit is decoded and the syndrome verified. One can decode directly or by reverse teleportation through the same type of Bell state used for the initial teleportation, verifying the syndrome in the teleportation process.…”

Temporal resources for global quantum computing architectures

“…For the other models analyzed here, QECC is possible as long as a degree of parallelism of at least O(log(N)) is feasible [34]. This fact makes the design of complementary strategies allowing parallel QECC processing in LM3 and BM1 a must.…”