High-Threshold Low-Overhead Fault-Tolerant Classical Computation and the Replacement of Measurements with Unitary Quantum Gates

Abstract

classic "multiplexing" method is unique in achieving high-threshold fault-tolerant classical computation (FTCC), but has several significant barriers to implementation: i) the extremely complex circuits required by randomized connections, ii) the difficulty of calculating its performance in practical regimes of both code size and logical error rate, and iii) the (perceived) need for large code sizes. Here we present numerical results indicating that the third assertion is false, and introduce a novel scheme that eliminates the two remaining problems while retaining a threshold very close to von Neumann's ideal of 1/6. We present a simple, highly ordered wiring structure that vastly reduces the circuit complexity, demonstrates that randomization is unnecessary, and provides a feasible method to calculate the performance. This in turn allows us to show that the scheme requires only moderate code sizes, vastly outperforms concatenation schemes, and under a standard error model a unitary implementation realizes universal FTCC with an accuracy threshold of p < 5.5%, in which p is the error probability for 3-qubit gates. FTCC is a key component in realizing measurement-free protocols for quantum information processing. In view of this we use our scheme to show that all-unitary quantum circuits can reproduce any measurement-based feedback process in which the asymptotic error probabilities for the measurement and feedback are (32/63)p ≈ 0.51p and 1.51p, respectively.PACS numbers: 03.67.-a, 03.67. Pp, 03.65.Ta, 03.65.Aa The problem of performing classical computing reliably with unreliable logic gates is referred to as faulttolerant classical computation (FTCC). The first method for realizing FTCC was devised by von Neumann, who called it multiplexing [1]. It achieves what may be the highest possible error threshold (the maximum stable component-wise error rate), but has hitherto been viewed as impracticable. This is due to an apparent need for high redundancy (number of fundamental components required to construct a noise-free logical gate), the need to continually connect and reconnect bits "at random" at a potentially large spatial separation, and the difficulty of both analytically calculating the performance for moderate code sizes and simulating the performance in the low-error regimes required for reliable computation [1][2][3][4][5][6][7][8][9]. The field of probabilistic cellular automata was partially motivated by addressing the second problem, but has not to-date produced a complete and feasible FTCC scheme [10][11][12][13][14][15][16][17][18][19][20][21]. A second method for FTCC was developed more recently in the context of quantum computing, and involves "concatenating" errorcorrection codes and logic gates . However, the concrete FTCC schemes developed using concatenation require high redundancy and connections between widely separated code bits, and have not to-date achieved the high thresholds of multiplexing schemes [47,48].We are interested in FTCC here primarily because of its central role in the ...

The role of quantum measurements in physical processes and protocols

“…This error probability for encoding is precisely equivalent to the measurement error in a measurementbased protocol. We recently presented an explicit scheme for encoding classical information and processing it reliably using unitary gates [20]. This method builds upon von Neumann's "multiplexing" method [21], appears to be quite feasible to implement (unlike previous multiplexing protocols), and achieves an error threshold close to von Neumann's ideal value of = 1/6 for a "majority organ" (see below) in an error-correcting network.…”

“…1 achieves the encoding process by copying the state of a single qubit to 3 n qubits by using a "cascade" of "AMP" gates, in which each AMP copies the classical information in a single qubit to two others (using, e.g., two CNOT's). The probability that there is an error in the resulting encoding is ≈ 0.51p, in which p is the probability that there is a (quantum) error in at least one of the outputs of the AMP gate [20]. Thus the equivalent measurement error incurred by this encoding circuit is ≈ 0.51p, and thus less than that of an individual AMP gate.…”

“…It turns out that it is possible to use majority gates in a manner that is much simpler and has a performance which is quite similar. By i) arranging the 3 n code qubits in an n-dimensional hypercube, ii) defining a single correction operation as that of sequentially applying majority gates along each of the n dimensions of the hypercube in turn, and iii) defining the code "bits" as the 3 n−1 groups containing three mutually correlated bits, one obtains a high-threshold fault-tolerant correction circuit that is highly efficient in that it enables a compact (wiring) configuration [20]. As an example, an 81-(qu)bit code (n=4) turns a three-bit unitary gate error of p = 0.4% into a logical error probability of 1.5×10 −11 .…”

“…In fact, the AMP and majority gates are universal for classical computation. Since all inputs of a threeinput gate must be correct in order to produce a correct logical result, the threshold is slightly smaller for universal computation and was calculated to be p = 5.5% [20] or ≈ 13%.…”

Quantum arithmetics via computation with minimized external control: The half-adder

“…On the other hand, we find a four-qubit quantum network that continuously implements a half-adder with a fidelity of ≈ 97.9%. Note that this is already better than the threshold for fault-tolerant classical computation using quantum gates [27].…”