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arXiv:2309.08663 · Physical Review A

Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code

Daniel Honciuc Menendez, Annie Ray, Michael Vasmer

Published Jun 28, 2024• 12 citations
arXivHTMLPDF

Abstract

Quantum computers promise to solve problems that are intractable for classical computers, but qubits are vulnerable to many sources of error, limiting the depth of the circuits that can be reliably executed on today's quantum hardware. Quantum error correction has been proposed as a solution to this problem, whereby quantum information is protected by encoding it into a quantum error-correcting code. But protecting quantum information is not enough, we must also process the information using logic gates that are robust to faults that occur during their execution. One method for processing information fault-tolerantly is to use quantum error-correcting codes that have logical gates with a tensor product structure (transversal gates), making them naturally fault-tolerant. Here, we test the performance of a code with such transversal gates, the [[8,3,2]] color code, using trapped-ion and superconducting hardware. We observe improved performance (compared to no encoding) for encoded circuits implementing non-Clifford gates, a class of gates that are essential for achieving universal quantum computing. In particular, we find improved performance for an encoded circuit implementing the controlled-controlled-$Z$ gate, a key gate in Shor's algorithm. Our results illustrate the potential of using codes with transversal gates to implement nontrivial algorithms on near-term quantum hardware.

DOI
10.1103/physreva.109.062438

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Quantum computers promise to solve problems that are intractable for classical computers, but qubits are vulnerable to many sources of error, limiting the depth of the circuits that can be reliably executed on today's quantum hardware. Quantum error correction has been proposed as a solution to this problem, whereby quantum information is protected by encoding it into a quantum error-correcting code. But protecting quantum information is not enough, we must also process the information using logic gates that are robust to faults that occur during their execution. One method for processing information fault-tolerantly is to use quantum error-correcting codes that have logical gates with a tensor product structure (transversal gates), making them naturally fault-tolerant. Here, we test the performance of a code with such transversal gates, the [[8,3,2]] color code, using trapped-ion and superconducting hardware. We observe improved performance (compared to no encoding) for encoded circuits implementing non-Clifford gates, a class of gates that are essential for achieving universal quantum computing. In particular, we find improved performance for an encoded circuit implementing the controlled-controlled-ZZZ gate, a key gate in Shor's algorithm. Our results illustrate the potential of using codes with transversal gates to implement nontrivial algorithms on near-term quantum hardware.
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.
We compare two different implementations of fault-tolerant entangling gates on logical qubits. In one instance, a twelve-qubit trapped-ion quantum computer is used to implement a non-transversal logical CNOT gate between two five qubit codes. The operation is evaluated with varying degrees of fault tolerance, which are provided by including quantum error correction circuit primitives known as flagging and pieceable fault tolerance. In the second instance, a twenty-qubit trapped-ion quantum computer is used to implement a transversal logical CNOT gate on two [[7,1,3]] color codes. The two codes were implemented on different but similar devices, and in both instances, all of the quantum error correction primitives, including the determination of corrections via decoding, are implemented during runtime using a classical compute environment that is tightly integrated with the quantum processor. For different combinations of the primitives, logical state fidelity measurements are made after applying the gate to different input states, providing bounds on the process fidelity. We find the highest fidelity operations with the color code, with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and 0.99959(13) when preparing eigenstates of the logical X and Z operators, which is higher than the average physical qubit SPAM fidelities of 0.9968(2) and 0.9970(1) for the physical X and Z bases, respectively. When combined with a logical transversal CNOT gate, we find the color code to perform the sequence--state preparation, CNOT, measure out--with an average fidelity bounded by [0.9957,0.9963]. The logical fidelity bounds are higher than the analogous physical-level fidelity bounds, which we find to be [0.9850,0.9903], reflecting multiple physical noise sources such as SPAM errors for two qubits, several single-qubit gates, a two-qubit gate and some amount of memory error.
Code switching is an established technique that facilitates a universal set of FT quantum gate operations by combining two QEC codes with complementary sets of gates, which each by themselves are easy to implement fault-tolerantly. In this work, we present a code switching scheme that respects the constraints of FT circuit design by only making use of transversal gates. These gates are intrinsically FT without additional qubit overhead. We analyze application of the scheme to low-distance color codes, which are suitable for operation in existing quantum processors, for instance based on trapped ions or neutral atoms. We briefly discuss connectivity constraints that arise for architectures based on superconducting qubits. Numerical simulations of circuit-level noise indicate that a logical T -gate, facilitated by our scheme, could outperform both flag-FT magic state injection protocols and a physical T -gate at low physical error rates. Transversal code switching naturally scales to code pairs of arbitrary code distance. We observe improved performance of a distance-5 protocol compared to both the distance-3 implementation and the physical gate for realistically attainable physical entangling gate error rates. We discuss how the scheme can be implemented with a large degree of parallelization, provided that logical auxiliary qubits can be prepared reliably enough. Our logical T -gate circumvents potentially costly magic state factories. The requirements to perform QEC and to achieve an FT universal gate set are then essentially the same: Prepare logical auxiliary qubits offline, execute transversal gates and perform fast-enough measurements. Transversal code switching thus serves to enable more practical hardware realizations of FT universal quantum computation. The scheme alleviates resource requirements for experimental demonstrations of quantum algorithms run on logical qubits.
Quantum error correction is a crucial tool for mitigating hardware errors in quantum computers by encoding logical information into multiple physical qubits. However, no single error-correcting code allows for an intrinsically fault-tolerant implementation of all the gates needed for universal quantum computing [1-3]. One way to tackle this problem is to switch between two suitable error-correcting codes, while preserving the encoded logical information, which in combination give access to a fault-tolerant universal gate set [4-6]. In this work, we present the first experimental implementation of fault-tolerant code switching between two codes. One is the seven-qubit color code [7], which features fault-tolerant CNOT and HHH quantum gates, while the other one, the 10-qubit code [8], allows for a fault-tolerant TTT-gate implementation. Together they form a complementary universal gate set. Building on essential code switching building blocks, we construct logical circuits and prepare 12 different logical states which are not accessible natively in a fault-tolerant way within a single code. Finally, we use code switching to entangle two logical qubits employing the full universal gate set in a single logical quantum circuit. Our results experimentally open up a new route towards deterministic control over logical qubits with low auxiliary qubit overhead, not relying on the probabilistic preparation of resource states.
Transversal logical gates provide a direct route to fault-tolerant quantum computation, but the Eastin-Knill theorem forbids a universal transversal gate set within a single quantum error-correcting code. We propose a hybrid architecture based on the tetrahedral three-dimensional color code and its Hadamard-transformed counterpart, which we call the H-tetrahedral code. The two encodings support complementary transversal non-Clifford operations. Combined with bitwise Hadamard transformations that switch between the two encodings and a one-way transversal logical CNOT from the tetrahedral code to the H-tetrahedral code, these operations realize an almost-universal transversal logical gate set that enables both the creation of entanglement and logical states with magic. We complete a universal gate set through a pieceably fault-tolerant round-robin construction of a logical controlled-ZZZ gate between two H-tetrahedral codes. This logical entangling gate is interleaved with reduced-overhead Steane-type syndrome extraction using logical two-dimensional color-code auxiliary qubits. Our construction provides a new route toward implementing classically hard-to-simulate quantum algorithms where magic and most entangling operations are transversal while the resource overhead is concentrated in a small number of non-transversal Clifford entangling operations.
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.