Experimental exploration of five-qubit quantum error correcting code with superconducting qubits

TL;DR

This paper reports the first experimental realization of the smallest five-qubit quantum error correcting code using superconducting qubits, demonstrating key features like error detection, logical operations, and state decoding.

Contribution

The work is the first to experimentally implement the $[ ext{!}[5,1,3] ext{!}]$ quantum error correction code with superconducting qubits, verifying all essential functionalities.

Findings

Encoded states with 57.1% fidelity

High fidelity of 98.6% within the code space

Successful error detection and state recovery

Abstract

Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this challenge, we experimentally realise the $[\![5,1,3]\!]$ code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the $[\![5,1,3]\!]$ code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\%$, in total with $92$ gates. Our work demonstrates each key aspect of the $[\![5,1,3]\!]$ code and verifies the viability of experimental realization of quantum error correcting codes with superconducting qubits.

Figures

Equations

$$g_1=X_1{Z}_2{Z}_3{X}_4{I}_5,$$

$$g_3=X_1{I}_2{X}_3{Z}_4{Z}_5,$$

$${\rho }_L=\frac{I+{\bar{P}}_X{X}_L+{\bar{P}}_Y{Y}_L+{\bar{P}}_Z{Z}_L}{2},$$

Full text

quantum error correcting code; superconducting qubit; five-qubit code; error detection; logical operation;

∗ These two authors contributed equally to this work.

† xma@tsinghua.edu.cn.

‡ yuaochen@ustc.edu.cn.

§ xbzhu16@ustc.edu.cn.

¶ pan@ustc.edu.cn.

I INTRODUCTION

Quantum computers can tackle classically intractable problems Shor94 and efficiently simulate many-body quantum systems lloyd1996universal . However, quantum computers are notoriously difficult to control, due to their ubiquitous yet inevitable interaction with their environment, together with imperfect manipulations that constitute the algorithm. The theory of fault tolerance has been developed as the long-term solution to this issue, enabling universal error-free quantum computing with noisy quantum hardware FaultTolerance ; Shorcode95 ; Steancode ; 5qubitcodetheory96 ; 5qubitTHeory296 . The logical qubits of an algorithm can be represented using a larger number of flawed physical qubits. Providing that the machine is sufficiently large (high qubit count), and that physical errors happen with a probability below a certain threshold, then such errors can be systematically detected and corrected knill2005quantum ; Threshold06 . In experiment, several small quantum error correcting codes (QECCs), including the repetition code chiaverini2004realization ; schindler2011experimental ; reed2012realization ; waldherr2014quantum ; riste2015detecting ; cramer2016repeated ; PhysRevA.97.052313 , the four qubit error detecting code lu2008experimental ; bell2014experimental ; linke2017fault , the seven qubit color code 7qubitCodeExp , the bosonic quantum error correcting code Ofek2016 ; Campagne-Ibarcq2020 , and others yao2012experimental ; taminiau2014universal ; kelly2015state ; Andersen2020 , have been realized with different hardware platforms. These works have shown the success of realising error correcting codes with non-destructive stabilizer measurements and its application in extending the system lifetime kelly2015state ; linke2017fault . Nevertheless, previous experiments are limited to restricted codes for correcting certain types of errors or the preparation of specific logical states. It remains an open challenge to realise a fully-functional QECC.

Here, we focus on the five-qubit $[\![5,1,3]\!]$ code, the ‘perfect’ code that can protect a logical qubit from an arbitrary single physical error using the smallest number of qubits 5qubitcodetheory96 ; 5qubitTHeory296 . While proof-of-principle experimental demonstrations of the $[\![5,1,3]\!]$ code have been conducted on NMR systems QEC98 , whether it could be incorporated in more scalable quantum computing systems and protect errors presented in these systems remain open. Here, we focus on the realization of the five-qubit code with superconducting qubit systems. As a preparatory theoretical step, we recompile the universal encoding circuit that prepares an arbitrary logical state in order to realise it with the fewest possible number of nearest-neighbour two-qubit gates. In experiment, we implement basic functions of the code by realizing logical states preparation, transversal logical operations, and state decoding.

II Theory

The five-qubit $[\![5,1,3]\!]$ code is a type of stabilizer code that is defined by a set of independent operators from the Pauli group, called stabilizers, such that the code space only has eigenvalue $+1$. The four stabilizers of the five-qubit code are

[TABLE]

with $I_{i}$, $X_{i}$, $Y_{i}$, $Z_{i}$ being the Pauli matrices acting on the $i^{\textrm{th}}$ qubit. The logical state space is defined by states $\ket{\Psi}_{L}=a\ket{0}_{L}+b\ket{1}_{L}$ that are simultaneously stabilized by the four stabilizers with $g_{i}\ket{\Psi}_{L}=+\ket{\Psi}_{L},\,\forall i=1,2,3,4$. Here, the logical states $\ket{0}_{L}$ and $\ket{1}_{L}$ are the basis states that are eigenstates of the logical $Z_{L}$ operator. Any logical state $\ket{\Psi}_{L}$ can be uniquely determined by the four stabilizers defined in Eq. (1) together with the fifth stabilizer $g_{5}=\ket{\Psi}\bra{\Psi}_{L}-\ket{\Psi^{\bot}}\bra{\Psi^{\bot}}_{L}=(aa^{\star}-bb^{\star}){Z}_{L}+(a^{\star}b+b^{\star}a){X}_{L}-i(a^{\star}b-b^{\star}a){Y}_{L}$, with $\ket{\Psi^{\bot}}=b^{\star}\ket{0_{L}}-a^{\star}\ket{1_{L}}$. That is, any logical state $\ket{\Psi}_{L}$ can be decomposed as $\ket{\Psi}_{L}\bra{\Psi}_{L}=2^{-5}\Pi_{i=1}^{5}(g_{0}+g_{i})$, where $g_{0}=I_{1}I_{2}I_{3}I_{4}I_{5}$ is the trivial stabilizer of all pure quantum states. Logical Pauli operators is transversely realised by applying the corresponding single-qubit gates on each physical qubit, $\sigma_{L}=\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}\sigma_{5}$ for $\sigma=X,Y,Z$. General logical operators, such as the $T_{L}=e^{-iZ_{L}\pi/8}$ gate, may not be transversely realised.

The five-qubit code has distance three and therefore all single-qubit errors can be identified (and thus corrected) while all double-qubit errors can be detected. When there is no error, all stabilizer measurements should yield $+1$ for the encoded state $\ket{\Psi}_{L}$. When an error happens, one or more stabilizer measurements may yield $-1$. As there are four stabilizers whose measurement may take either $+1$ or $-1$ values, there are in total $15$ syndrome measurement results with at least one outcome being $-1$. If we consider the ways in which a single Pauli error can afflict one of the five qubits, we note that there are fifteen possibilities (three error types and five locations), with each mapping to a specific one of the fifteen syndromes. When a two-qubit error happens, we again find that at least one of the stabilizer measurements takes $-1$. This heralds the fact that some error has occurred. However, since different double-qubit errors may have the same syndrome, we can only detect double-qubit errors without the capability of identifying or correcting them. Nevertheless, this latter property can be useful in some situations, such as state preparation, where we can simply discard a faulty realisation and restart.

Without using ancillary qubits, the original circuit for encoding the logical state $\ket{\Psi}_{L}$ requires a number of two qubit gates which are non-local with respect to a linear architecture 5qubitcodetheory96 ; 5qubitTHeory296 . To tailor the circuit for superconducting systems that only involve nearest-neighbour controlled-phase gates, we recompile the encoding circuit to have the minimal possible number (eight) of nearest-neighbour control phase gates as shown in Fig. 1(a). We leave the details of circuit compilation in Supplementary Data.

III Experimental set-up

The device for the implementation of the five-qubit error correcting code is a 12-qubit superconducting quantum processor Gong2018 . Among these 12 qubits, we chose five adjacent qubits to perform the experiment. The qubits are capacitively coupled to their nearest neighbours. The capacitively coupled $XY$ control lines enable the application of single-qubit rotation gates by applying microwave pulses, and the inductively coupled $Z$ control lines enable the double-qubit controlled-phase gates by adiabatically tune the two-qubit state $\ket{11}$ close to the avoid level crossing of $\ket{11}$ and $\ket{02}$ Gong2018 . After careful calibrations and gate optimizations, we have the average gate fidelities as high as $0.9993$ for single-qubit gates and $0.986$ for two-qubit gates. With the implementation of only single-qubit rotation gates and double-qubit controlled-phase gates, we realized the circuit for encoding and decoding of the logical state. More details about the experimental set-up are shown in Supplementary Data.

IV Results

On a superconducting quantum processor Gong2018 , we experimentally realised the logical states $\ket{0}_{L}$, $\ket{1}_{L}$, $\ket{\pm}_{L}$, and $\ket{\pm i}_{L}$ that are eigenstates of the logical Pauli operators $X_{L}$, $Y_{L}$, and $Z_{L}$, and the magic state $\ket{T}_{L}=(\ket{0}_{L}+e^{i\pi/4}\ket{1}_{L})/\sqrt{2}$ that cannot be realized by applying Clifford operations on any eigenstate of the logical Pauli operators. The expectation values of the stabilizer operators of $\ket{T}_{L}$ are shown in Fig. 1(b). The fidelity between the experimentally prepared state and the ideal state $\ket{\Psi}_{L}\bra{\Psi}_{L}$ is determined by the measurement of the $32$ stabilizer operators in $\Pi_{i=1}^{5}(g_{0}+g_{i})$. We omit the $g_{0}$ one as it is constantly 1. In this way, we obtained the state fidelity as the average of the 32 stabilizers by picking up corresponding measurement results among the state tomography results. Finally, the state fidelity of $\ket{T}_{L}$ reaches 54.5(4)%. The fidelities of other prepared states are shown in Supplementary Data, with an average fidelity being $57.1(3)\%$. The main error in preparing the encoded state is from decoherence, especially the relatively short dephasing time. In a numerical simulation of the experiment with decoherence (see Supplementary Data for details), the state fidelity of $\ket{T}_{L}$ is $58.9\%$. After numerically increasing the dephasing time to be the same as the energy relaxation time, the state fidelity can be increased to $92.1\%$, indicating a potential direction for future improvements.

The quality of the prepared logical states can also be divided into its overlap with the logical code space and its agreement with the target logical state after projecting it into the code space. Given the logical Pauli operators $X_{L}$, $Y_{L}$, $Z_{L}$ and $I_{L}=\ket{0}_{L}\bra{0}_{L}+\ket{1}_{L}\bra{1}_{L}$, the normalised density matrix $\rho_{L}$ is defined by projecting the experimentally prepared state $\rho_{q}$ into the code space

[TABLE]

with normalised probability $\bar{P}_{j}=P_{j}/P_{I}$ and $P_{j}=\textrm{Tr}(\rho_{q}\sigma_{L})$, for all $\sigma=I$, $X$, $Y$, $Z$, where $\rho_{q}$ is the density matrix of the five-qubit state. We define the fidelity within the code space by $\mathcal{F}_{L}=\bra{\Psi}_{L}\rho_{L}\ket{\Psi}_{L}$, as shown in Fig. 1(c), with the average value being as high as $98.6(1)\%$. Since projecting to the code space is equivalent to post-selecting all $+1$ stabilizer measurements, our result also indicates the possibility of high fidelity logical state preparation with future non-destructive stabilizer measurements. This relies on whether we can achieve accurate non-destructive stabilizer measurements, especially whether errors from the ancillary qubits and additional gates are sufficiently low.

Given the realisation of logical state, one proceed with the verification of error correction/detection ability of the five qubit code. Acting on the logical encoded state $\ket{T}_{L}$, we systematically introduce every type of single-qubit error by artificially applying the corresponding single qubit gate on one of the five qubits. Then, by measuring the four stabilizers $g_{1}$, $g_{2}$, $g_{3}$, and $g_{4}$, we aim to verify that each error would be properly identified. As shown in Fig. 2(a) we do indeed find, for each case, the corresponding syndrome pattern that identifies the location of the single-qubit error. Suppose the expectation value of $i$-th stabilizer is $p_{i}$, the probability that the syndrome measurement works is $\prod_{i}(|p_{i}|+1)/2$, which is 0.413 on average in experiment. We also apply double-qubit errors on $\ket{T}_{L}$ and find the same syndrome correlation that can always detect the existence of errors (see Supplementary Data for details). Notably, the (single-qubit or double-qubit) error-afflicted states have probabilities projecting onto the code space (around $3.3\%$), verifying the power of the error correcting code.

In a functioning fault-tolerant quantum computer, operations on logical qubits are realised through a series of operations on the component physical qubits. We implement and verify three such transversal logical operations: Starting from the magic state $\ket{T}_{L}$ presented in Fig. 3(a), we demonstrate the single logical qubit operations $X_{L}$, $Y_{L}$, and $Z_{L}$, and plot the rotated states within the code space, as shown in Fig. 3(b), (c), and (d), respectively. To characterize these logical operations, we performed the quantum process tomography within the code space as shown in Figure 3(e), which reflects how well logical operations manipulate logical states. We determine gate fidelities of the logical $X_{L}$, $Y_{L}$, and $Z_{L}$ operations to be $97.2(2)\%$, $97.8(2)\%$, and $97.3(2)\%$, respectively.

Finally, after encoding the single-qubit input state into the logical state, we apply the decoding circuit, see Fig. 4(a), to map it back to the input state. With input states $\ket{0}$, $\ket{1}$, $\ket{+}$, and $\ket{+i}$, we determine the state fidelity after decoding as 87.4(5)%, 91.6(4)%, 76.7(6)%, and 77.1(6)%, respectively. The relatively lower fidelities for $\ket{+}$ and $\ket{+i}$ states are also caused by the short dephasing time. After quantum process tomography from the four output states, the process fidelity is determined as $74.5(6)\%$ as shown in Fig. 4(b). The decoding circuit only apply operations on three qubits, highlighting the ability of quantum secret sharing with the five-qubit code PhysRevLett.83.648 . This simplification owes to a consequence of locality: no observable on Q1 can be affected by the omitted independent gate operations of the other qubits.

V Discussion

An essential milestone on the road to fault-tolerant quantum computing is the achievement of error-corrected logical qubits that genuinely benefit from error correction, outperforming simple physical qubits. There are three steps for achieving this goal — (1) realizing encoded logical qubits in a code capable of detecting and correcting errors, (2) realizing operations on encoded qubits and error correction cycles, and (3) adding more ancillary qubits and improving the operation fidelity to achieve fault-tolerance. Our experiment completes step (1) by realising the basic ingredients of the full functional five-qubit error correcting code. Our work partially achieves (2) as we indeed perform logical operations and verify error detection; however because we are only able to evaluate stabilizers destructively, we cannot perform full error correction. Direction for future works include the the realization of non-destructive error detection kelly2015state ; Bultinkeaay3050 ; Andersen2020 and error correction, and the implementation of logical operations on multiple logical qubits for the five-qubit code. Our work also has applications in error mitigation for near-term quantum computing mcclean2020decoding .

DATA AVAILABILITY STATEMENT

All data analyzed to evaluate the conclusions are available from the authors upon reasonable request.

Acknowledgments

The authors thank the USTC Center for Micro- and Nanoscale Research and Fabrication, Institute of Physics CAS and National Center for Nanoscience and Technology for the support of the sample fabrication. The authors also thank QuantumCTek Co., Ltd. for supporting the fabrication and the maintenance of room temperature electronics.

Funding

This work was supported by the National Key Research and Development Program of China (2017YFA0304300, 2017YFA0303900 and 2017YFA0304004), the National Natural Science Foundation of China (11875173, 11674193, 11574380 and 11905217), the Chinese Academy of Science, Science and Technology Committee of Shanghai Municipality (16DZ2260100), Anhui Initiative in Quantum Information Technologies, and Engineering and Physical Sciences Research Council (EP/M013243/1 to S.C.B and X.Y.).

Author contributions

X. Ma, Y.-A. Chen, X.-B. Zhu, and J.-W. Pan conceived the research. M. Gong, X. Yuan, X. Ma, and X.-B. Zhu designed the experiment. S.-Y. Wang designed the sample. H. Deng and H. Rong prepared the sample. X. Yuan, Z. Zhang, Q. Zhao, Y.-C. Liu, and H. Lu designed the quantum circuit. M. Gong, Y.-L. Wu, Y.-W. Zhao, C. Zha, and S.-W. Li carried out the experiments. Y.-L. Wu developed the programming platform for measurements. M. Gong, X. Yuan, Y.-W. Zhao, C. Zha, S. Benjamin, X. Ma, Y.-A. Chen, and X.-B. Zhu analyzed the results. F.-T. Liang, J. Lin, Y. Xu, and C.-Z. Peng developed room temperature electronics equipments. All authors contributed to discussions of the results and development of manuscript. X.-B. Zhu and J.-W. Pan supervised the whole project.

Conflict of interest statement: None declared.