EError-transparent operations on a logical qubit protected by quantum error correction
Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, ∗ and L. Sun
1, † Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
Universal quantum computation [1] is striking for its unprecedented capability in processing information, butits scalability is challenging in practice because of the inevitable environment noise. Although quantum errorcorrection (QEC) techniques [2–7] have been developed to protect stored quantum information from leadingorders of errors, the noise-resilient processing of the QEC-protected quantum information is highly demandedbut remains elusive [8]. Here, we demonstrate phase gate operations on a logical qubit encoded in a bosonicoscillator in an error-transparent (ET) manner. Inspired by Refs. [9, 10], the ET gates are extended to thebosonic code and are able to tolerate errors during the gate operations, regardless of the random occurrencetime of the error. With precisely designed gate Hamiltonians through photon-number-resolved AC-Stark shifts,the ET condition is fulfilled experimentally. We verify that the ET gates outperform the non-ET gates with asubstantial improvement of the gate fidelity after an occurrence of the single-photon-loss error. Our ET gates inthe superconducting quantum circuits are readily for extending to multiple encoded qubits and a universal gateset is within reach, paving the way towards fault-tolerant quantum computation.
The uncontrollable noise in a quantum system is the mostsignificant obstacle in realizing universal quantum computa-tion [1], since the induced errors are unpredictable and dele-terious to the encoded quantum information. Quantum errorcorrection (QEC) is proposed to tackle this problem [11] byexpanding the dimension of the Hilbert space for quantum in-formation and thus introducing the redundancy to tolerate theleading errors. In conventional QEC, quantum informationis encoded on logical qubits, constructed by multiple physi-cal qubits, within a subspace spanned by the QEC codewordscalled the code space. Although each physical qubit is sus-ceptible to noise, errors can be detected without corruptingthe stored quantum information while mapping the quantumstate in the code space to the orthogonal error spaces. Overthe past years, great progress has been achieved in QEC the-ories, and proof-of-principle demonstrations of error detec-tion and correction are reported in various experimental plat-forms [2–7]. Especially, the break-even point of QEC hasbeen demonstrated with a logical qubit encoded in a bosonicoscillator [12].However, QEC can merely maintain the stored quantumstates from noise. Errors occurring during the execution ofquantum operations might accumulate and spread over thequantum circuits, so that the processing of information is notreliable. Fault-tolerant universal quantum computation archi-tectures [8], such as the transversal gates on logical qubitsand magic-state distillation, were developed for performingnoise-resilient quantum gates on encoded qubits, but the im-plementations are extremely challenging. Instead of realiz-ing a complete fault-tolerant architecture, practical schemesthat demonstrate the key ideas in a near-term few-qubit sys-tem were proposed [13, 14]. Only very recently, fault-tolerantstate preparation [15] and error detection [16, 17] were ex-perimentally demonstrated. An alternative approach of fault-tolerant operations based on the concept of error-transparent(ET) gates [9, 10] was proposed theoretically and promisesfault-tolerant non-Clifford logical gates. Nevertheless, its im-plementation in the multi-qubit QEC codes requires many-
Not Error Transparent E rr o r S pa c e C ode S pa c e Error TransparentHilbert Space ˆ E ˆ E Hilbert Space ( ) ˆ L U t ( ) L ψ ( ) L T ψ ba FIG. 1.
Concept of error-transparent (ET) gate.
During a quantumgate on a logical state with finite operation time, error E might occurrandomly, mapping the state in the code space (upper gray cube) tothe orthogonal error space (lower purple cube). For the non-ET gate( a ), the track of state evolution in the error space is different for E occurring at different time. When the ET condition is satisfied ( b ),the tracks in both the code and error spaces are deterministic andidentical whenever the error occurs. Therefore, E can be detectedand corrected with QEC after the gate operation, making the ET gatefault tolerant. body interactions and is hard to realize experimentally.Here, we extend the concept of ET gates to bosonic QECcodes and experimentally demonstrate ET arbitrary phasegates that tolerate the single-photon-loss error. The ET gatesare successfully validated by the remarkable improvement ofthe coherence of the logical states after the occurrence of anerror during the evolution of the gates. By applying repet-itive autonomous QEC (AQEC), the ET gates on the QEC-protected logical qubits show higher fidelities than the cases a r X i v : . [ qu a n t - ph ] S e p with non-ET gates or without AQEC. Our results promise auniversal ET gate set for quantum computation, thus revealsthe potential of the bosonic quantum computation architec-ture [18, 19], and presents a first step towards the fault-tolerantquantum computation.The basic idea of the ET operation on a logical qubit isillustrated in Fig. 1. Applying a Hamiltonian H ( t ) to the log-ical qubit, any encoded quantum state is expected to evolveas | ψ L ( t ) i → U ( t , t ) | ψ L ( t ) i with a target unitary opera-tion U ( t , t ) = T e − i ´ t t d τ H ( τ ) ( T is the time-ordering oper-ator). However, because errors could occur during the oper-ation process, the logical state will consequently jump fromthe code space to the error space. Due to the stochastic na-ture of noise, the practical evolution of the state may fol-low different tracks, as shown in Fig. 1 a . For example, ifan error E j occurs at time t , the track leads to the final state | e ψ ( t ) i = U ( T , t ) E j U ( t , ) | ψ L ( ) i in the error space. An EToperation requires a deterministic track of the logical stateevolution irrespective to t , as shown in Fig. 1 b , and the tar-get operation could always be achieved by mapping the stateback to the code space after the operation.Therefore, we derive the condition for theET operation as U ( T , t ) E j U ( t , ) | ψ L ( ) i = e i φ ( t ) E j U ( T , ) | ψ L ( ) i , ∀ j , t , | ψ L ( ) i by considering thefact that a global phase φ ( t ) makes no influence on the logicalstate. Note that the ET condition discussed in Refs. [9, 10]is a more restricted case of our condition with φ ( t ) = P † j H ( t ) P j = P †C H ( t ) P C + c ( t ) P C , ∀ j , t , (1)where P C is the projector onto the code space, P j ∝ E j P C is the projector from the code space to the error space corre-sponding to the error E j , and c ( t ) is a complex number.To demonstrate the ET operations on a bosonic logicalqubit, we explore a superconducting circuit consisting of ahigh-quality microwave cavity constituting the bosonic log-ical qubit and a dispersively coupled transmon qubit as theancilla [20, 21], and the system Hamiltonian reads H = ∆ ω a † a − χ a † a | e i h e | − K a †2 a . (2)Here, ∆ ω is the cavity frequency with respect to a carefullychosen local oscillator reference, a † ( a ) is the creation (an-nihilation) operator for the bosonic mode, | e i ( | g i ) is the ex-cited (ground) state of the ancilla, and χ / π = .
60 MHz and K / π = .
80 kHz are the dispersive coupling and Kerr coeffi-cients originated from the ancilla, respectively. To correct thedominant photon-loss errors in the bosonic mode, we encodethe quantum information on the lowest-order binomial code inthe cavity [22, 23], which is defined in Fock basis as | L i = | i + | i√ , | L i = | i . (3) We note that proper reference frame needs to be carefully cho-sen such that there is no accumulation of the relative phase be-tween Fock states | i and | i . When a single-photon-loss erroroccurs, the quantum state jumps into the error space spannedby the basis states | E i = | i , | E i = | i . (4)When prepare the logical qubit in the code space and set theancilla to the idle state | g i , a phase operation on the logicalqubit can be easily realized via the Kerr effect since P †C H P C = K ( I − Z ) (5)with respect to the code basis states (Eq. 3). Here, I and Z arethe Pauli matrices. Thus, an arbitrary phase gate R Kerr ( φ ) = e i φ Z on a single logical qubit can be implemented by waitingfor a duration of τ = φ / K . However, such phase operationscan not tolerate single-photon-loss errors, because the ET con-dition is not satisfied as P †E H P E = KI . (6)Furthermore, the cavity’s Kerr nonlinearity associated withthe coupling to the ancilla cannot be switched off, thereforethe stored logical qubit is always impacted by R Kerr and suf-fers random photon-loss-error-induced dephasing [23].To meet the ET condition, we develop a technique to flex-ibly engineer the Hamiltonian in both the code and errorspaces. Through a detuned microwave drive on the ancilla,photon-number-resolved AC-Stark shift (PASS) can be real-ized. As schematically shown in Fig. 2 a , due to the strongancilla-cavity dispersive coupling, the transition frequency ofthe ancilla is photon-number ( n ) dependent, and thus an off-resonant drive would induce photon-number dependent en-ergy shift δ n due to the AC-Stark effect [24, 25]. Such a fre-quency shift can also be understood as a geometric phase ac-cumulation ∼ δ n τ = Ω ∆ d − n χ τ for the joint ancilla-cavity state | gn i (Fig. 2 b ), while keeping the excitation to | en i negligibledue to the large detuning (Supplementary Information). Here τ is the gate duration time, Ω is the Rabi drive frequency, and ∆ d is the drive detuning with respect to the ancilla transitionfrequency corresponding to n =
0. By applying drives withcarefully chosen frequencies and amplitudes, we could pre-cisely engineer the frequency shifts of the Fock states to real-ize the Hamiltonian H PASS = n trc ∑ n = δ n | n ih n | , (7)with the truncated photon number n trc = c shows themeasured Fock state frequencies when applying a continuousdrive in the middle of the ancilla transition frequencies corre-sponding to n = n =
4, i.e. ∆ d = − . χ (the dashedorange line in Fig. 2 a ). The experimental results are well con-sistent with the theoretical predictions. I m ( α ) -2 2 Re ( α ) a ⟩ | ⟩ | ⟩ | 3 ⟩ | 3 ⟩ | 2 ⟩ | 2 ⟩ | 1 ⟩ | 1 ⟩ | 0 ⟩ | 0 ⟩ |0 ⟩ | Parity MeasurementError Detec � on ( ) ParityMeasurementWigner Tomography /2 E n c o d e ⟩ | ⟩ | bcd e R Kerr gate R ET gate I ET gate30 µ s 30 µ s 30 µ s60 µ s 60 µ s 60 µ s90 µ s 90 µ s 90 µ s-80-60-40-200204060 F r equen cy S h i ft ( k H z ) Ω/2π (MHz) -2 220-2 -2 220-220-2-2 220-2 -2 220-220-2-2 220-2 -2 220-220-2
FIG. 2.
Photon-number-resolved AC-Stark shift (PASS) and ET phase gates. a,
Illustration of the energy diagram in the strongly disper-sively coupled bosonic mode-ancilla system. A detuned microwave drive on the ancilla would induce PASS for each transition frequency. b, Geometric phase interpretation of the PASS. Fast rotating of the state in the Bloch space {| gn i , | en i} induces a phase accumulation propor-tional to time, equivalent to an energy shift of Fock state | n i . c, Measured frequencies of Fock states with respect to | i as a function of theamplitude of a continuous drive. The drive frequency is in the middle of the ancilla transition frequencies corresponding to Fock states | i and | i , as indicated by the horizontal dashed orange line in a . These results fit excellently with the theoretical predictions (dashed lines). Thevertical arrow indicates the amplitude to realize the ET phase gate R ET . d, Experimental sequence to characterize the ET gates. After R Kerr , R ET , or I ET , an error detection measurement is performed, followed by a Wigner tomography. e, Evolution of the logical state encoded withthe lowest-order binomial code ( | L i − i | L i ) / √ µ s, 60 µ s, and 90 µ s. Each leftcolumn is the Wigner function for a detection of no error, while each right one for a detection of an error. For non-ET R Kerr , the state after anerror eventually loses the phase information and becomes a mixed state; while for the ET gates, the coherence of the state in the error space ispreserved. All the Wigner functions are experimentally measured ones.
After experimentally validating the precisely controlledPASS, we turn to realize the ET phase gate R ET on the log-ical qubit. With an appropriate drive amplitude, we can obtain P †C ( H + H PASS ) P C = K ( I − Z ) (8)and P †E ( H + H PASS ) P E = K ( I − Z ) + cI , (9)with K / π = .
33 kHz and c / π = − .
63 kHz. Here, theET condition is satisfied with a re-chosen reference ∆ ω = .
09 kHz. We now verify the ET property of the phasegate by measuring the evolution of a logical state ( | L i − i | L i ) / √ R ET ) and without ( R Kerr ) the PASS. To sep-arately check the quantum evolution in the code and errorspaces, we perform Wigner tomography of the output statesby post-selecting the parity of the excitation number after var-ious evolution times (Fig. 2 d ) with the assistance of the an-cilla [23, 26]. The results are summarized in Fig. 2 e . Com-paring R ET and R Kerr , the evolution of the logical state in the code space shows similar rotations and phase coherence forboth cases, as indicated by the fringes in the azimuth direc-tion. However, the phase coherence in the error space is onlypreserved by R ET , in strong contrast to the significant corrup-tion of phase coherence for R Kerr , manifesting the tolerance tothe stochastic photon-loss error during the ET gate.Additionally, the ET idle gate I ET can also be realized bysimultaneously applying two PASS drives, such that K = e , I ET showssimilar ET properties as R ET , while the phase of the logicalstate remains unchanged in both the code and error spaces.To demonstrate the potential of the ET gates for fault-tolerant quantum computation, we further investigate the ETgates under AQEC protection, with the experimental sequenceshown in Fig. 3 a . An AQEC pulse numerically optimized witha duration of 1 . µ s recovers the error state during the ET gateoperation and also transfers the error entropy associated with P r o c e ss F i de li t y F T G ( µ s) No QEC-3.0-2.0-1.00.0 P ha s e / π T G ( µ s) code space error space code space error space b c d R Kerr R ET I ET I ET R ET R Kerr R Kerr R Kerr R ET R ET P r o c e ss F i de li t y F T G ( µ s) ⟩ |0 ⟩ | E n c o d e D e c o d e /2/2 A Q E C P u l s e Autonomous QEC ⟩ | /2/2 - . ( ) k H z - . ( ) k H z a FIG. 3.
ET gates on a logical qubit with autonomous quantum error correction (AQEC). a,
Experimental procedure for the ET gateperformance characterization. The gate operation on the logical qubit is realized by the corresponding Hamiltonian with varying gate time T G ,and a QEC process is implemented before the decoding. The QEC process consists of an AQEC pulse followed by an ancilla measurementand a reset. b, Phase shift of the logical state extracted from the process tomography. During the phase gates, the phases of the logical statechange linearly with T G . c, The gray dotted curve is the process fidelity F of R Kerr without AQEC and could be regarded as a reference. TheAQEC indeed improves F except for T G being small. The ET gates perform better than R Kerr as expected. d, F in the error and code spacesrespectively. In the code space, both R Kerr and R ET have nearly identical F . However, in the error space R ET has much higher F than R Kerr ,corroborating that the ET gate is able to protect the state when an error occurs during the gate. Note that at T G = µ s) ( ) µ s ( ) µ s ( ) µ s ( ) µ s No QEC R Kerr R ET I ET P r o c e ss F i de li t y F b ⟩ |0 ⟩ | A Q E C A Q E C … E n c o d e D e c o d e /2/2 /2/2 a FIG. 4.
ET gates protected by repetitive AQEC. a,
Experimentalsequence. b, Experimental process fidelity F as a function of timewith repetitive and interleaved ET gates and AQECs on the logicalqubit. Both ET gates have better performance than the non-ET R Kerr .The lifetime with the ET idle gate I ET is shorter than that with the ETphase gate R ET because of extra ancilla excitation from the additionaldrive. All these three gates with AQEC show better performance thanthe case without AQEC. the logical state to the ancilla, followed by a measurement-based ancilla reset (Methods and Supplementary Informa-tion). The AQEC is equivalent to previously demonstratedfeedback-based QEC [23], but holds the advantages of conve-nience in experiments and avoids the latency in the electroniccontrol system since the AQEC is error-detection free [3, 4, 6].Figures 3 b-d summarize the experimental results and pro-cess fidelities with different gate operation time T G . Arbitraryphase gates can be achieved with appropriate T G , howevertheir gate fidelities F decay with T G as expected. In Fig. 3 c ,we find all the gates are improved by AQEC when comparedwith R Kerr without AQEC, while the ET gates show superiorperformances. By measuring the process fidelity F in the codeand error spaces separately, the ET effect is clearly evidencedin Fig. 3 d : F for R Kerr and R ET in the code space are almostidentical, but F for R ET in the error space is substantially im-proved.Finally, the ET logical gate can be interleaved with AQECand performed repeatedly, as illustrated in Fig. 4 a . Figure 4 b shows the measured process fidelity decaying exponentially asa function of time. We have chosen the optimal time intervalfor each gate (60 µ s for R Kerr and 120 µ s for the ET gates).Clearly, both ET gates have better performance than the non-ET R Kerr . The lifetime with I ET is shorter than that with R ET because of extra ancilla excitation from the additional drive,which causes dephasing of the logical state. In addition, allthese three gates with AQEC have better performance thanthe case without AQEC, demonstrating the effectiveness ofAQEC.We introduce the concept of ET gates on a bosonic logicalqubit, where the evolution in the error space is independentand exactly the same as that in the code space. The ET arbi-trary phase gates and the idle gate have been demonstrated onthe lowest-order binomial code by engineering the frequencyshift of each Fock state, and an enhancement on the ET gate fi-delity has also been demonstrated with repetitive AQEC. Ourapproach could also be generalized to single-qubit Hadamardgate and controlled-phase gate on two binomial logical qubits(Supplementary Information), thus constituting the universalET gate set for quantum computation. 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This work was supported by National Key Research and De-velopment Program of China (Grant No.2017YFA0304303)and the National Natural Science Foundation of China(Grant No.11474177 and 11874235). C.-L.Z. was supported by National Natural Science Foundation of China (GrantNo.11874342 and 11922411) and Anhui Initiative in Quan-tum Information Technologies (AHY130200). upplementary Information for “Error-transparent operations on a logical qubit protected byquantum error correction”
Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, ∗ and L. Sun
1, † Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
I. EXPERIMENTAL PARAMETERS AND TECHNIQUESA. Experimental device
The experimental device is composed of two high-qualitythree-dimensional coaxial aluminum cavities [1–3] ( S and S ), three ancillary transmon qubits ( Q , Q , Q ) and threestripline cavities [4] ( R , R , R ), as shown in Fig. S1. Thedetailed geometry of the device can be found in Ref. [5]. Forcurrent experiment on the error-transparent (ET) gates on asingle-logical qubit, we only use the left part of the device ( S , Q , and R ), and the remaining parts stay in their ground statesduring the experiment. The single-logical qubit is encoded inthe bosonic mode of S (referred as the ‘cavity’ henceforth),which is dispersively coupled to the ancilla Q . The striplinecavity R with a high external coupling rate ( κ out ) is to readout Q . The relevant parameters and coherence properties of thedevice under study are listed in Table S1 and Table S2. Notethat the coefficients χ = χ qs and K = K s are used in the mainmanuscript and below for abbreviation. B. Experimental techniques
In this work, we have developed two experimental tech-niques to realize the ET gates under the protection of repeti-tive quantum error correction (QEC). The first technique is thephoton-number-resolved AC-Stark shift (PASS) to engineerthe system Hamiltonian precisely, and thus the ET condition Q R S Q R S Q R FIG. S1.
Experimental device.
The ET gate experiment is based onthe parts in the dashed rectangle, consisting of a high-quality three-dimensional coaxial aluminum cavity S as the storage cavity to en-code the logical qubit, an ancillary transmon qubit Q , and a striplinecavity R for readout of the ancilla. The remaining parts are in theirground states during the experiment. Term Value ω q / π ω s / π ω r / π K q / π
252 MHz K s / π χ qs / π χ qr / π Experimentally characterized parameters for thecavities and the ancilla qubit.
Here, the subscripts (q,s,r) denotethe ancilla qubit ( Q ), storage cavity ( S ) and readout cavity ( R ),respectively. ω is the bare frequency for each compoment, K is theself-Kerr coefficient and χ is the dispersive coupling strength. Q Q S R T µ s 25 µ s 480 µ s 58 ns T µ s 30 µ s 560 µ s - T φ µ s 75 µ s 1.3 ms - n th .
6% 0 . <
1% -TABLE S2.
Coherence properties of the ancilla qubits and thecavities. T and T are the experimentally measured energy andphase relaxation times, respectively, T φ is the derived pure dephasingtime, and n th is the thermal excitation in the experiment. can be satisfied for the binomial codes. The second techniqueis the autonomous QEC (AQEC), by which the single-photon-loss error can be detected and corrected without extracting theerror syndrome by the control electronics and thus the elec-tronic latency can be avoided. In this section, we provide thedetails of the two techniques.
1. PASS
For the system under study, the Hamiltonian for realizingPASS reads H = ∆ ω a † a − K a †2 a + (cid:0) ω q − χ a † a − ω d (cid:1) | e i h e | + Ω ( | e i h g | + | g i h e | ) , (S1)where ∆ ω is the cavity frequency with respect to a carefullychosen local oscillator reference, a † ( a ) is the creation (an-nihilation) operator for the bosonic mode, | e i ( | g i ) is the ex-cited (ground) state of the ancilla, χ is the dispersive couplingstrength, K is the self-Kerr coefficient of the cavity originated a r X i v : . [ qu a n t - ph ] S e p gmem engn } ∆ θ n ∆ } m ∆ Ω Θ Ω d ∆ FIG. S2.
Geometric phase interpretation of the PASS.
An off-resonant drive with a frequency ∆ d between two dispersive transitionfrequencies of the ancilla and an amplitude Ω would produce a ge-ometric phase on each ancilla state. This phase is accumulated con-stantly and causes an equivalent frequency shift on the photon Fockstate because the ancilla is approximately in its ground state and canbe traced out. The direction of the frequency shift is related to thesign of the detuning ∆ . from the ancilla, ω q is the ancilla qubit frequency when thecavity is in vacuum, ω d is the driving frequency on the an-cilla, and Ω is the Rabi drive frequency. In the following, wedefine ∆ d ≡ ω q − ω d . In the limit of Ω (cid:28) (cid:12)(cid:12) ∆ d − χ a † a (cid:12)(cid:12) for allcavity states, we have the effective Hamiltonian as [6] H eff ≈ ∆ ω a † a − K a †2 a + (cid:0) ∆ d − χ a † a (cid:1) | e i h e | + Ω ( ∆ d − χ a † a ) ( | e i h e | − | g i h g | ) . (S2)Then, the effective frequencies for | gn i and | en i (the jointancilla-cavity states) are n ∆ ω − n ( n − ) K − Ω ( ∆ d − n χ ) and n ∆ ω − n ( n − ) K +( ∆ d − n χ )+ Ω ( ∆ d − n χ ) , respectively. There-fore, the effective frequency for Fock state | n i is shifted by ± Ω ( ∆ d − n χ ) conditional on n and the state of the ancilla qubit.As a result, the detuned drive induces the PASS. For ex-ample, for the drive frequency lying between ancilla transi-tion frequencies corresponding to | n i and | n + i , i.e. ω q − ( n + ) χ < ω d < ω q − n χ , the PASS on | gn i and | g ( n + ) i are negative and positive, respectively.The PASS can also be understood from the point view ofthe geometric phase accumulated on the photonic Fock statedue to the off-resonant drive, as shown in Fig. S2. The ancillastate initialized at the pole cannot be efficiently excited, butonly rotates near the pole. The solid angle enclosed by thetrajectory of the state on the Bloch sphere can be representedas Θ = π [ − cos ( θ )] = π ( − ∆ √ Ω + ∆ ) . (S3)For the detuned drive, the cycle period for the state rotating on the Bloch sphere is T cyc = π √ Ω + ∆ , (S4)so the accumulated geometric phase ( Θ /
2) is proportional tothe number of rotation cycles, and the equivalent frequencyshift can be derived as δ ( ε , ∆ ) = Θ T cyc = √ Ω + ∆ − ∆ ≈ Ω ∆ (S6)for | ∆ / Ω | (cid:29) κ q ), and for the ancilla preparedin the ground state, the effective Hamiltonian becomes H eff ≈ ∆ ω a † a − K a †2 a − Ω ( ∆ d − χ a † a ) − i κ q . (S7)Therefore, the PASS drive would not only induce the energylevel shift H PASS = ∑ n δ n | n ih n | = ∑ n Ω ∆ d − n χ | n ih n | , (S8)but also induce the phase decoherence of the Fock states witha rate γ n ≈ Ω ( ∆ d − n χ ) κ q . (S9)This equation indicates that the frequency shift can be used toimplement a logical-qubit phase gate while the error broughtfrom the ancilla is significantly suppressed because of the an-cilla’s small excitation during the gate.Here, we also want to briefly discuss the limitation of thePASS technique. For the purpose of error transparency in thiswork, the PASS should compensate the self-Kerr effect, i.e. δ n = O ( K ) . If an individual PASS drive is applied to | gn i and | g ( n + ) i , we would have ∆ d − n χ ≈ χ , and thus the compen-sation requires Ω χ = O ( K ) . In addition, the PASS requiresa small drive amplitude Ω (cid:28) χ and negligible induced deco-herence γ n (cid:28) n κ a , with κ a being the amplitude decay rate ofthe cavity. From Eq. (S9), we have γ n ≈ Ω χ κ q χ = κ q χ O ( K ) .Taking the fact that the self-Kerr coefficient of the cavityis related to the cross-Kerr coefficient (dispersive couplingstrength) as K = χ / E c , with E c being the anharmonicity(the self-Kerr) of the ancillary transmon qubit, we can derivethe conditions for the PASS drive to achieve the ET gates as:(1) Ω = χ O (cid:16)q χ E c (cid:17) , (2) q χ E c (cid:28)
1, and (3) χ E c (cid:28) κ a κ q .For the device in this study, we have E c / π ∼
252 MHz, χ / π ∼ .
60 MHz and κ a κ q ∼ .
07, therefore χ E c ∼ . × − and all the above conditions are satisfied. For a better perfor-mance of the PASS technique for ET gates, κ q and χ E c of thesuperconducting circuit should be further reduced.3
2. AQEC
AQEC is equivalent to the standard measurement-basedQEC, which consists of both error detection and correctionoperations. However, AQEC does not need error detections.It is worth noting that the AQEC had been used with the three-qubit repetition code [7–9], while its extension to the bosoniccodes requires rather sophisticated conditional unitary opera-tions. To perform AQEC for the bosonic codes, a unitary tran-sition is implemented to correct the logical state in the errorspace | ψ E i while driving the ancilla to an orthogonal state: U | ψ E i| g i = | ψ L i| e i , (S10)but keep the logical state in the code space | ψ L i unchanged: U | ψ L i| g i = | ψ L i| g i . (S11)In this case, the correlation between the quantum system andthe environment (which induces errors) is erased, and thus theerror entropy is transferred to the ancilla. After the operation,the logical state is recovered, and the ancilla system can betraced out. Note that in our experiment | ψ E i = a | ψ L i (single-photon-loss error) in the error space; while there is a non-unitary no-jump evolution e − κ a a † at / of | ψ L i in the code space,which is corrected in the AQEC pulse.In practice, we need to reuse the ancilla, therefore we resetthe ancilla to a pure state after each implementation of AQECfor the next round of AQEC. Compared with previous demon-strations of QEC [10], the whole process does not need anyprojective measurement on the encoded bosonic state, and theerror syndrome is not necessarily to be extracted. Therefore,the real-time feedback control system is not required any moreand the potential electronic latency is avoided. The reset of theancilla could be implemented in either digital or analog ap-proaches. For the digital approach, the ancilla can be directlyreadout, and a control pulse dependent on the readout resultis then applied to reset the ancilla. In this measurement-basedcase, the digital control only needs to implement the reset be-fore the next AQEC step, which is hundreds of microsecondslater in our case, in contrast to a few hundred nanosecondsrequired for real-time feedback control. For the analog ap-proach, the ancilla could be engineered to couple to a readoutcavity by switching on a stimulating drive, which can allowthe decay of the excitation in the ancilla within about one mi-crosecond. Here in this experiment, we use the so-called gra-dient ascent pulse engineering (GRAPE) algorithm [11, 12] tonumerically optimize the AQEC pulse and use measurement-feedback method to reset the ancilla. II. MORE EXPERIMENTAL DATAA. Detailed experimental data of the PASS
In the experiment, we have used one or two microwavedrives to precisely control the energy shifts of the photon num-ber states, according to the photon-number-resolved AC-Stark P opu l a t i on P opu l a t i on µ s) , | 〉 〉 R Kerr R ET I ET R Kerr R ET I ET FIG. S3.
Ramsey-type experiments in the code and error spaces.
The Ramsey experiment is performed with a superposition of Fockstates | i and | i (the error space basis); and Fock states | i and | i (the code space basis). The synchronized oscillations with bothone and two PASS drives in the two spaces demonstrate the satis-faction of the ET condition. However, when the PASS drive is off,the oscillations are not synchronized any more, indicating the non-ET condition. The dashed lines are fits with a decayed sinusoidalfunction. effect. The experimental technique and theoretical details areprovided in the next section. Here, we show extra experimen-tal results complementary to the presented ones in the maintext.Table S3 summarizes the parameters of the microwavedrives used in the experiment. ∆ d is the drive detuning with re-spect to the ancilla qubit transition frequency when the cavityis in a vacuum state (i.e. there is no photon-induced frequencyshift to the ancilla qubit), and Ω is the corresponding Rabidrive frequency that is proportional to the microwave drivingamplitude. Here, all the driving parameters are optimized tominimize the excitation of the ancilla qubit in the simulation,and then carefully calibrated in the experiment.For the ET phase gate, there is only one microwave drivewith ∆ d being between − χ and − χ . Therefore, this drivemakes the frequency shifts of Fock states | i and | i in op-posite directions, thus satisfying the ET condition ( f − f ) − ( f − f ) =
0. Here f n are measured through Ramsey-type ex-periments on superposition states ( | i + | n i ) / √ ∆ ω = f n for the three different gates R Kerr , R ET , and I ET are also provided in Table S3. We findthat the ET condition is indeed satisfied. In comparison, thecase without the PASS drive (the phase gate R Kerr due to theKerr coefficient) has ( f − f ) − ( f − f ) ≈ π ×
10 kHz.For the ET gate R ET , we choose a cavity frequency in areference frame with ∆ ω / π = .
09 kHz to fix the phaseof Fock state | i , and then there is a non-zero rotating fre-quency of Fock state | i relative to Fock states | i and | i ,i.e. f / − f = .
67 kHz. As a result, a phase gate can berealized to the binomial code.By adding one more PASS drive with ∆ d being between4 Driving parameter ET phase gate ET IdleAmplitude Ω . χ . χ , . χ Detuning ∆ d − . χ − . χ , − . χ Fock state frequency shift (2 π kHz) | i | i | i | i f − f f − f f / − f phase gate due to Kerr R Kerr . ( ) − . ( ) − . ( ) − . ( ) − . − . − . R ET − . ( ) − . ( ) − . ( ) − . ( ) − . − . − . I ET − . ( ) − . ( ) − . ( ) − . ( ) − . − . − . Parameters of the PASS drives and frequency shifts in the experiment.
The numbers in the parenthesis are the measurementuncertainty. P r o c e ss F i de li t y F T G ( µ s) 0.90.80.70.60.5 Code space Error space Probability 0.90.80.70.60.50.40.3 3002001000 T G ( µ s) 0.90.80.70.60.5 N o - e rr o r p r obab ili t y Code space Error space Probability0.90.80.70.60.50.40.3 3002001000 T G ( µ s) 0.90.80.70.60.5 Code space Error space Probability Total Total Total R ET gate I ET gate R Kerr gate
FIG. S4.
Process fidelity of the three single-logical-qubit gates with AQEC in the error space and code space respectively.
The dashedlines corresponding to the right vertical axis are the probabilities of detecting no-error (no single-photon loss). − χ and − χ , we can further compensate the relative rotatingfrequency between | L i = ( | i + | i ) / √ | L i = | i , andessentially generate the ET idle gate I ET in the same referenceframe as R ET . With the additional PASS drive, the inducedfrequency shifts of Fock states | i and | i have different di-rections. From Table S3, we find that ( f − f ) − ( f − f ) = f / − f = B. Process fidelities for the single-logical-qubit gates withAQEC
Figure S4 provides more concrete data for the three differ-ent gates, accompanying Fig. 3 in the main text. The perfor- mances of the ET and non-ET gates are characterized by mea-suring the process fidelity F as a function of the gate time T G .Here, F is separately measured for the code and error spacesby post-selecting the ancilla state that indicates if an error hap-pens or not. The experimentally measured probabilities of no-error happening are also provided, by which the total fidelitycan be derived as a weighted combination of those in both theerror and code spaces.The main feature for the three gates in Figure S4 is thatthe fidelities in the error space for the ET gates R ET and I ET are very similar and much higher than the non-ET gate R Kerr .Such a difference manifests that the ET gates possess the capa-bility of protecting the quantum information in the error spacefrom corruption.The no-error probability decays with T G as expected, andso does the fidelity in the code space for all these three gates.However, the fidelities in the error space are low for T G = T G is that inthese cases the error is mainly induced by the ancilla deco-herence, ancilla excitation, and operation errors, instead ofthe single-photon-loss error (has not happened yet) that theET gates can protect. When T G is large enough, the fidelityin the error space becomes much higher because the con-tribution of single-photon-loss errors dominates in the errorspace. When T G further increases, the uncorrectable high-5 P r o c e ss F i de li t y F T G ( µ s) total code space error space 3002001000 T G ( µ s) total code space error space1.0000.990 2010 1.0000.990 2010 3002001000 T G ( µ s) total code space error space1.0000.990 2010 Idle without Kerr Idle with Kerr R ET b ca FIG. S5.
Numerical simulation results with ideal AQEC. a,
The ideal idle gate with no cavity’s self-Kerr effect. b, The idle gate withcavity’s self-Kerr effect. The logical state in the code space can still preserve, but corrupts quickly in the error space because the ET conditionis not satisfied. c, ET phase gate with the PASS drive. The main trend is the same as a when T G is sufficiently large. However, there is a littleloss when T G approaches zero because the effective Hamiltonian is no longer satisfied in this regime. T G ( µ s) R ET code space error space1.00.90.80.70.60.50.40.3 P r o c e ss F i de li t y F T G ( µ s) R Kerr code space error space1.00.90.80.70.60.50.4 P r o c e ss F i de li t y F T G ( µ s) R ET with ancilla thermal excitation code space error space R ET with ancilla thermal excitation R Kerr with detailed simulation R ET with detailed simulation a b FIG. S6.
Numerical simulation results with ancilla thermal excitation and operation errors. a,
The ET phase gate with the ancilla thermalexcitation, but still with ideal AQEC. The fidelity decays more linearly, in contrast to the quadratically decay curves in Fig. S5 c with zero bathtemperature. b, More detailed simulation of the non-ET phase gate R Kerr and the ET phase gate R ET including all possible imperfections. Theresults in b are very similar to the measured ones in Fig. 3 c of the main text. order photon-loss errors happen with higher probabilities, andtherefore the overall fidelities in both the code and error spacesdecay.For the non-ET gate, the fidelity decays much faster be-cause of the fast dephasing of quantum information in the er-ror space due to the non-ET Kerr effect. Although the non-ETgate loses the phase information in the error space quickly, itsfidelity remains at about 0 . III. NUMERICAL ANALYSIS
To analyze sources of errors in the experiment and study theviability of the ET gates for potential fault-tolerant quantumcomputation, we implement numerical simulations accordingto our experimentally calibrated parameters. It is anticipatedthat the main experimental imperfections include: (1) the im-perfect AQEC operations on the system, (2) the decoherenceof the ancilla qubit during the ET gates, and (3) the thermalexcitation of the ancilla qubit. In the following, we comparethe results under different situations with the noiseless idealcase.6
A. Numerical simulation with ideal AQEC
First of all, we use the ideal AQEC process in the numericalsimulation to study the viability of the PASS technique for therealization of ET phase gates. The decoherence (damping anddephasing) of both the ancilla and the cavity are included inthe numerical model. However, we assume the thermal bathare in the vacuum state, so the ancilla cannot be populated bythe thermal noise from the bath. By substituting the experi-mentally calibrated parameters into the master equations, thesystem evolutions are numerically solved and the correspond-ing process fidelities are summarized in Fig. S5. Similar tothe experiments in the main text, the ancilla qubit is traced outafter the ET gate evolution with different gate times, and thedensity matrices in the error and code spaces are separatelyprocessed to obtain the process fidelities.Comparing the results from the ET and non-ET gates inFig. S5, we find the performances of the ET gates (Fig. S5 c )are close to the ideal case without the cavity’s self-Kerr ef-fect (Fig. S5 a ). No self-Kerr effect of the cavity is a pre-assumption for many theoretical works on quantum gates [13].However, when the self-Kerr presents in the cavity, the idleoperation does not satisfy the ET condition, so the state in theerror space corrupts quickly (Fig. S5 b ).By applying the PASS technique on the system to engineerthe Hamiltonian, the ET condition can be satisfied and theprocess fidelity in the error space preserves. However, wefind that the curves of the ET phase gates are slightly lowerthan the ideal case. The reason could be attributed to theexcitation of the ancilla by the off-resonant drive. As pre-dicted by Eq. S9, the state jump of the ancilla qubit wouldinduce the dephasing of the cavity states. In addition, there isa small loss of fidelity when T G approaches zero (Fig S5 c in-set). This is because the effective Hamiltonian approximationfor PASS is only satisfied when the gate time is sufficientlylong T G (cid:29) / ∆ . Therefore, for T G < µ s in our experiment,the geometric phase cannot be regarded as continuously accu-mulated, and the PASS is deviated from Eq. S8. B. Numerical simulation with ancilla thermal excitation andimperfect AQEC
The above simulations reveal the effect of the imperfectionsdue to the drive-induced ancilla excitation and the consequentancilla-excitation-induced dephasing, as predicted in the pre-vious section on PASS. In practice, there are more imperfec-tions that could induce the loss of gate fidelity.One main contribution to the loss is the ancilla thermal ex-citation. The results including the ancilla thermal bath areshown in Fig. S6 a . The fidelity decays more linearly, in con-trast to the quadratically decay curves in Fig. S5 c with zerobath temperature. As shown in Fig. S7, the ancilla excita-tion caused by the PASS drive increases with the drive ampli-tude. For the parameter used in our ET experiments ( Ω / π ∼ . E xc i t a t i on Ω /2 π (MHz) ⟩ |0 FIG. S7.
Ancilla excitation from the PASS drive.
The ancillaexcitation due to an off-resonant drive with a detuning frequency ∆ d = − . χ are measured as a function of the drive amplitude whenthe cavity is initialized at differnet Fock states. Dotted lines are sim-ulation results. The ET gate used in the experiment is performed withthe drive amplitude Ω / π ≈ . .
01, while the thermal exci-tation is about 0 . the measured thermal excitation in the experiment. Therefore,we conclude that the photon dephasing is mainly from the an-cilla thermal excitation, which can be suppressed by a coldbath. The PASS-drive-induced dephasing could be potentiallysolved by using alternative bosonic codes that are robust todephasing errors, such as the cat code [14, 15] or the numeri-cally optimized codes [16–18].In the experiment, there are inevitable imperfections in theAQEC pulse and the measurement-feedback operation. To ac-count for these errors on the performance of the non-ET gate R Kerr and the ET phase gate R ET , a more detailed simulationincluding these imperfections are carried out, and the resultsare shown in Fig. S6 b . The obtained results agree well withthe experimental results in Fig. 3 of the main text, indicatingthe main imperfections in our experiments are due to the ther-mal excitation of the ancilla qubit and the operation errors. IV. THEORYA. Requirement for error transparency
For an ET evolution, an error occurring at a random instant t should not affect the final output state except for an extraglobal phase, which can be represented by: U ( T , t ) E j U ( t , ) | ψ L i = e i φ ( t ) E j U ( T , ) | ψ L i , ∀ i , j , t . (S12)Here, | ψ L i is an arbitrary logical quantum state in the codespace, and E j is in the error set. Because Eq. S12 should besatisfied for arbitrary time t , it is equivalent to U ( t + δ t , t ) E j | ψ L i = e i δφ ( t ) E j U ( t + δ t , t ) | ψ L i . (S13)7As a property of the logical gate, U ( t + δ t , t ) cannot introduceleakage out of the code space, i.e. U ( t + δ t , t ) = P C U ( t + δ t , t ) P C with P C being the projector onto the code space.Then the condition Eq. S13 can be transformed as U ( t + δ t , t ) E j P C | ψ L i = e i δφ ( t ) E j P C U ( t + δ t , t ) P C | ψ L i , (S14)by adding the projectors. By combining the requirements forQEC P C E † j E j P C = α j P C , α j ∈ R , (S15)the ET condition becomes P C E † j U ( t + δ t , t ) E j P C = e i δφ ( t ) α j P C U ( t + δ t , t ) P C . (S16)After introducing the projector P j = √ α j E j P C from thecode space to the error space due to E j , the above equationbecomes P † j U ( t + δ t , t ) P j = e i δφ ( t ) P C U ( t + δ t , t ) P C . (S17)Since U ( t + δ t , t ) = I − iH ( t ) δ t + O ( δ t ) , the ET conditioncan also be represented by the system Hamiltonian as P † j H ( t ) P j = P C H ( t ) P C + c ( t ) P C , (S18)where c ( t ) = − d φ ( t ) / dt . Here, the Hamiltonian is presentedin the code space.In previous theoretical works [19, 20], the condition of ETgates is derived as [ E j , H ( t )] | ψ L i i = , ∀ i , j , t . (S19)Comparing with the ET condition (Eq. S18) derived above, thecommutation relation is too strict, and Eq. S19 is just equiva-lent to a special case of Eq. S18 with φ ( t ) =
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