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arXiv:2412.04791

Fault-Tolerant Implementation of the Deutsch-Jozsa Algorithm

Singh, Divyanshu, Shiroman Prakash

Published Dec 6, 2024• 0 citations
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Abstract

We show that one can implement the Deutsch-Josza algorithm, one of the first and simplest quantum algorithms, in a fault-tolerant manner using the smallest quantum error-detecting code -- the $[[4,2,2]]$ code -- without any ancillae. We implemented the algorithm on a trapped-ion quantum computer with and without fault-tolerant encoding and compared the results. With approximately $99 \%$ confidence, we found that the fault-tolerant implementation provided a noise reduction for all oracles. Averaged across all oracles, the reduction in error rate was nearly $90 \%$.

DOI
10.48550/arxiv.2412.04791

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We show that one can implement the Deutsch-Josza algorithm, one of the first and simplest quantum algorithms, in a fault-tolerant manner using the smallest quantum error-detecting code -- the [[4,2,2]][[4,2,2]][[4,2,2]] code -- without any ancillae. We implemented the algorithm on a trapped-ion quantum computer with and without fault-tolerant encoding and compared the results. With approximately 99%99 \%99% confidence, we found that the fault-tolerant implementation provided a noise reduction for all oracles. Averaged across all oracles, the reduction in error rate was nearly 90%90 \%90%.
Many current quantum error-correcting codes that achieve full fault tolerance suffer from having low ratios of logical to physical qubits and significant overhead. This makes them difficult to implement on current noisy intermediate-scale quantum (NISQ) computers and results in the inability to perform quantum algorithms at useful scales with near-term quantum processors. As a result, calculations are generally done without encoding. We propose a middle ground between these two approaches: constructions in the [[n,n−2,2]][[n,n-2,2]][[n,n−2,2]] quantum error-detecting code that can detect any error from a single faulty gate by measuring the stabilizer generators of the code and additional ancillas at the end of the computation. This achieves weak fault tolerance. As we show, this yields a significant improvement over no error correction for small computations with low enough physical error probabilities and requires much less overhead than codes that achieve full fault tolerance. We give constructions for a set of gates that achieve universal quantum computation in this error-detecting code, while satisfying weak fault tolerance up to analog imprecision on the physical rotation gate.
Fault-tolerant operations based on stabilizer codes are the state of the art in suppressing error rates in quantum computations. Most such codes do not permit a straightforward implementation of non-Clifford logical operations, which are necessary to define a universal gate set. As a result, implementations of these operations must use either error-correcting codes with more complicated error correction procedures or gate teleportation and magic states, which are prepared at the logical level, increasing overhead to a degree that precludes near-term implementation. Here, we implement a small quantum algorithm, one-qubit addition, fault-tolerantly on a trapped-ion quantum computer, using the [[8, 3, 2]] color code. By removing unnecessary error correction circuits and using low-overhead techniques for fault-tolerant preparation and measurement, we reduce the number of error-prone two-qubit gates and measurements to 36. We observe arithmetic errors with a rate of ∼1.1 × 10−3for the fault-tolerant circuit and ∼9.5 × 10−3for the unencoded circuit.
A fully fault-tolerant implementation of the quantum error-detecting Iceberg [[2m,2m−2,2]][[2m, 2m-2, 2]][[2m,2m−2,2]] code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.