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Acta Physica Sinica

Fault-tolerantly implementing dense rotation operations based on non-stabilizer states

Wu Xiang-Yan, Xu Yan-Ling, Yu Ya-Fei, Zhiming Zhang

Published Jan 1, 2014• 0 citations
Publisher

Abstract

Based on the quantum error-correction codes and concatenation, quantum logical gates can be implemented transversally, which is called the fault-tolerant quantum computation. Clifford gates can be directly and fault-tolerantly performed, but they cannot reach universal quantum computation. How to implement the non-Clifford gate fault-tolerantly is a vital technique in fault-tolerant universal quantum computation. Here the magic state is selected to help the implementing of the non-Clifford gate transversally. Based on the non-stabilizer state cos θi|0>+sinθi|1>, circuits which can execute 2θi rotation around X-axis and Z-axis fault-tolerantly are proposed. Then new non-stabilizer states in this form are developed and produced from the distilled magic state. By using these states, a number of non-Clifford gates can be performed transversally, which makes profound implication in fault-tolerant quantum computation. We calculate the number of the non-stabilizer states needed for simulating the desired rotation operations, which is less than that in previous protocols.

DOI
10.7498/aps.63.220304

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Based on the quantum error-correction codes and concatenation, quantum logical gates can be implemented transversally, which is called the fault-tolerant quantum computation. Clifford gates can be directly and fault-tolerantly performed, but they cannot reach universal quantum computation. How to implement the non-Clifford gate fault-tolerantly is a vital technique in fault-tolerant universal quantum computation. Here the magic state is selected to help the implementing of the non-Clifford gate transversally. Based on the non-stabilizer state cos θi|0>+sinθi|1>, circuits which can execute 2θi rotation around X-axis and Z-axis fault-tolerantly are proposed. Then new non-stabilizer states in this form are developed and produced from the distilled magic state. By using these states, a number of non-Clifford gates can be performed transversally, which makes profound implication in fault-tolerant quantum computation. We calculate the number of the non-stabilizer states needed for simulating the desired rotation operations, which is less than that in previous protocols.
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.
Abstract Magic states have been widely studied in recent years as resource states that help quantum computers achieve fault-tolerant universal quantum computing. The fault-tolerant quantum computing requires fault-tolerant implementation of a set of universal logical gates. Stabilizer code, as a commonly used error correcting code with good properties, can apply the Clifford gates transversally which is fault tolerant. But only Clifford gates cannot realize universal computation. Magic states are introduced to construct non-Clifford gates that combine with Clifford operations to achieve universal quantum computing. Since the preparation of quantum states is inevitably accompanied by noise, preparing the magic state with high fidelity and low overhead is the crucial problem to realizing universal quantum computation. In this paper, we survey the related literature in the past 20 years and introduce the common types of magic states, the protocols to obtain high-fidelity magic states, and overhead analysis for these protocols. Finally, we discuss the future directions of this field.
Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.
We introduce a class of three-dimensional color codes, which we call stacked codes, together with a fault-tolerant transformation that will map logical qubits encoded in two-dimensional (2D) color codes into stacked codes and back. The stacked code allows for the transversal implementation of a non-Clifford $\ensuremath{\pi}/8$ logical gate, which when combined with the logical Clifford gates that are transversal in the 2D color code give a gate set that is both fault-tolerant and universal without requiring nonstabilizer magic states. We then show that the layers forming the stacked code can be unfolded and arranged in a 2D layout. As only Clifford gates can be implemented transversally for 2D topological stabilizer codes, a nonlocal operation must be incorporated in order to allow for this transversal application of a non-Clifford gate. Our code achieves this operation through the transformation from a 2D color code to the unfolded stacked code induced by measuring only geometrically local stabilizers and gauge operators within the bulk of 2D color codes together with a nonlocal operator that has support on a one-dimensional boundary between such 2D codes. We believe that this proposed method to implement the nonlocal operation is a realistic one for 2D stabilizer layouts and would be beneficial in avoiding the large overheads caused by magic state distillation.
Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity 7−1+3×10−57^{+3}_{-1}\times 10^{-5}7−1+3​×10−5 with a 14.8−1+1%14.8^{+1}_{-1}\%14.8−1+1​% discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity ≤2.3−9+9×10−4\leq 2.3^{+9}_{-9}\times 10^{-4}≤2.3−9+9​×10−4. Notably, this significantly outperforms the unencoded physical CH infidelity of 10−310^{-3}10−3. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of 6×10−106\times 10^{-10}6×10−10 (5×10−145\times 10^{-14}5×10−14) at two-qubit error rate of 10−310^{-3}10−3 (10−410^{-4}10−4) respectively.
Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of magic gate teleportation (MGT), a form of quantum gate teleportation that uses non-stabilizer resource states to implement non-Clifford unitary gates on arbitrary input states without revealing information about those states. This setting is motivated by the canonical approach to implementing logical non-Clifford gates in fault-tolerant quantum computation. We uncover a hidden structure within MGT---after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that directly useful single-qubit resource states are necessarily Clifford-equivalent to diagonal states, and establish the same constraint for multi-qubit resource states used in a broad class of MGT protocols. In particular, the output state distilled from the [ ⁣[5,1,3] ⁣][\![5, 1, 3]\!][[5,1,3]] protocol is not directly useful for any MGT protocols, though it can be probabilistically converted into directly useful states. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.