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Certified quantum gatesWesley C. Campbell11University of California Los Angeles(Dated: July 9, 2020)High quality, fully-programmable quantum processors are available with small numbers ( 1000)of qubits, and the scientific potential of these near term machines is not well understood. If thesmall number of physical qubits precludes practical quantum error correction, how can these errorsusceptible processors be used to perform useful tasks? We present a strategy for developing quantum error detection for certain gate imperfections that utilizes additional internal states and doesnot require additional physical qubits. Examples for adding error detection are provided for a universal gate set in the trapped ion platform. Error detection can be used to certify individual gateoperations against certain errors, and the irreversible nature of the detection allows a result of acomplex computation to be checked at the end for error flags.For the near-term future, it is likely that the quantum information processors that become available willbe capable of running intermediate scale algorithms inthe presence of multiple (possibly numerous) errors [1].For beyond-classical computations this paradigm, the result reported by the quantum computer is almost guaranteed to be wrong, and the recent observation of quantum advantage by the Google group [2] was made possible only by arguing that after repeating the algorithmmany times, the algorithmic error probability could bemade statistically distinguishable from 1. For algorithmswhere the result can be tested directly for correctness(such as Shor’s factoring algorithm [3]), this may be useful, at least up to the point where the ratio of the runtime to success probability exceeds practical timescales.However, for many applications of quantum computers(such as the sampling problem used to demonstrate quantum advantage, and much of quantum simulation), theuser has very little idea which results are the trustworthyones, potentially rendering any purported quantum advantage effectively useless. Quantum advantage is likelynecessary, but not generally sufficient, to realize quantumutility beyond classical machines.Here, we consider the issue of how to deal with errorsin quantum processors caused by imperfections in the applied gates. While the techniques we outline below areapplicable to other hardware platforms, we present themin the context of trapped ion hyperfine qubits, which areeffectively free of errors outside of those caused by thegates themselves. In particular, since frequency stabilityis typically easier to distribute, assess, and achieve thanamplitude stability, errors caused by frequency drifts areusually unlikely compared to errors in the areas of pulsesapplied to perform gates, and we therefore focus primarily on amplitude errors. Composite pulse sequences [4]can be used to suppress amplitude (and frequency) errors that are common mode for the duration of a composite pulse sequence, but do not perform well againstcorrelated errors that are not constant during the sequence, such as amplitude drifts from amplifier temperature changes or laser intensity noise. Far from beingexotic or implausibly insidious, these types of amplitudedrift errors, which degrade the protection afforded bycomposite pulse sequences, have posed obstacles for anumber of experiments working at the forefront of fidelity[5–7].In this paper, we present a strategy for designing certifiable gates that uses auxiliary states in each qubit hostand does not require additional physical qubits. Thelarger Hilbert space afforded by including ancillary statesallows us to re-structure a gate as a series of population transfer steps that are each followed by dissipationof the error state through coupling to a bath/detector.Specifically, each step is designed as a rotation from theinitial state ψn i to an orthogonal target state ψn 1 i.By choosing ψn i and ψn 1 i to reside in orthogonalHilbert spaces, this rotation can be attempted and certified without acquiring knowledge of the informationencoded in either state. If the execution of this rotation is imperfect due to an error in the degree of therotation (i.e. the amplitude), the system will be left in φn 1 i ψn 1 i ψn i, and subsequent detection thatthe system is not in the error state ψn i certifies the stepagainst the rotation error. Since the dissipative detectionstep is irreversible, testing for errors can be done eitherduring the computation or at the end, and checking aresult for error flags can serve as a limited test of thetrustworthiness of the result.We begin with an example that illustrates the mainidea in the form of arbitrary single-qubit gates that theuser can certify against single-pulse amplitude errors.Examples of how to certify against errors in multi qubitcontexts such as addressing errors and 2-qubit entanglinggate errors are also presented, demonstrating that a complete set of gates for universal quantum computing canbe augmented with certification against some classes oferrors.We consider a system consisting of a qubit ( 0i and 1i) and two additional long-lived auxiliary states A( ) iand A(-) i that can each be coupled to both qubit statesvia resonant radiation. For concreteness, we will suppose that the qubit and auxiliary states are encoded in

2the effect of the gate toF 4F 3m -1ΘΘU (n̂, Θ) ψ0 i e i 2 c( ) ni ei 2 c( ) ni.f1 f2f3m 0f4m 1m 2oFIG. 1. Example 2F7/2-state encoding of the qubit and auxil171 iary states inYb . For storage, the qubit can be encodedin the two clock states and then transferred to and from thisarrangement before and after gates. All four transitions arewithin Zeeman shifts of the zero-field hyperfine splitting of3.602(2) GHz [8].oZeeman sub-levels of the effectively stable 2F7/2state of 171Yb , shown in Fig. 1. The qubit states can be defined as 0i F, MF i 3, 0i and 1i 3, 1i, and theauxiliary states as A( ) i 4, 0i and A(-) i 4, 1i. Astable, static magnetic field provides the qubit splitting,and the qubit and auxiliary manifolds are separated byothe 2F7/2hyperfine splitting ( 3.6 GHz; we will referto the coupling fields as microwaves). Further, we require that the system possess a means by which projective quantum measurement can be performed selectivelyfor population in each of these two manifolds. In thisexample, detection can be effected by hyperfine-selectivetransfer to the ground 2S1/2 state via optical pumping atoλ 760 nm on 1[3/2]o3/2 2F7/2, followed by spontaneousemission on 1[3/2]o3/2 2S1/2 . We have confirmed experimentally that this measurement can be accomplished ina few milliseconds with greater than 95% hyperfine manifold selectivity [9], and the theoretical limit is greaterthan 1 10 5 .Without loss of generality, we adopt a state vectordescription of the gate operation for clarity. Beforewe describe the certified gate protocol, we can consider the action of a general, unitary, single-qubit gateU (n̂, Θ) exp ( i Θ n̂·σ/2) on an arbitrary pure inputstate ψ0 i α 0i β 1i. If we rewrite the initial state inthe basis of ni (the eigenvectors of n̂ · σ), we have ψ0 i c( ) ni c( ) ni,(1)where c( ) h n ψ0 i. The states ni can likewise bewritten in terms of the polar (θ) and azimuthal (φ) anglesof n̂ on the Bloch sphere as θθiφ 0i e sin 1i ni cos22 θθ ni sin 0i eiφ cos 1i.(2)22This choice of basis simplifies the expression describing(3)For a certifiable version of the gate U (n̂, Θ), first, amicrowave pulse with four simultaneous tones (fi , seeFig. 1) transfers (ideally all) the population from thequbit states to the auxiliary states according to ni A( ) i. Each of the ni basis states is paired with onlyone of the auxilliary states A( ) i by two of the fourtones fi and acts as a coherent dark state with respectto the other two. The relative phases (ϕ12 and ϕ34 ) andRabi frequencies (Ωi ) chosen for the four frequencies depend only on the angles used to describe n̂, φ and θ(respectively, see Eq. (2)). Specifically, ϕ12 ϕ3,4 φ,Ω1 Ω4 Ω cos(θ/2), and Ω2 Ω3 Ω sin(θ/2). Inthe rotating frame with respect to the four splittings, theinteraction Hamiltonian is Ω2 iφΩ1h0 e h1 H A( ) i22 Ω3Ω4 i(φ π) A( ) ih0 eh1 H.c. (4)22 Ω ( ) A ih n A( ) ih n H.c.(5)2where we assume the splittings are such that the fourfrequencies are non-degenerate.Since these four sinusoids can be generated by a singlesynthesizer (for instance, a digital arbitrary waveformgenerator utilizing a single voltage reference) and canbe made to share a single transmission system, amplifierchain, antenna, etc., we consider the case in which theamplitude error of this step is a fractional amplitude error that is shared by all four coupling terms. Since weseek full transfer from the qubit manifoldRto the auxiliarymanifold, we represent the pulse area as dtΩ π δπn ,where δπn is the result of an amplitude error for the nthstep of the gate. We can write the state of the systemafter the (possibly imperfect) transfer asδπ1δπ1) (c( ) A( ) i c(-) A(-) i) sin() ψ0 i22(6)which is in the desired form for error detection,p φn 1 i 1 2 ψn 1 i ψn i,(7) φ1 i i cos(if we identify the error as sin(δπ1 /2).Next, any population left in the qubit manifold ( 0i and 1i, see Fig. 1) is dissipatively transferred to 2S1/2 via optical pumping. This “clean out” process will be accompanied by subsequent fluorescence detection of ground-statepopulation at some point – right away or potentially evenup until very end of an algorithm. If the ion is queriedimmediately, it will yield fluorescence (a “bright stateion”) with small probability sin2 (δπ1 /2). If the ion isnot in the bright state, the dissipative process has completed the successful transfer of all qubit population to

3the auxiliary manifold, yielding the desired target statefree of that error, ψ1 i i (c( ) A( ) i c(-) A(-) i).For the third step, a second pulse with the same fourtones is applied to transfer (ideally all) the populationfrom the auxiliary manifold back to the qubit manifold.The only difference between the waveform for the firstand second pulses is that a common phase shift π Θ/2is added to tones f1 and f2 only, and a common phaseshift of π Θ/2 is added to tones f3 and f4 only. Againkeeping track of a potential (possibly different) amplitudeerror that gives rise to finite δπ2 in the nominal π-pulse,the system is left in Θδπ2 i Θ ( )) e 2 c ni ei 2 c(-) ni φ2 i cos(2δπ2 sin() ψ1 i,(8)2which is in form (7) for sin(δπ2 /2).As the final step, any population left in the auxiliarymanifold ( A( ) i, viz. ψ1 i) is optically pumped to theground state, either yielding a bright state (with probability sin2 (δπ2 /2)) or completing the transfer to produceΘΘ ψ2 i U (n̂, Θ) ψ0 i e i 2 c( ) ni ei 2 c(-) ni, (9)the ideal gate with no contribution from the amplitudeerrors.The gate protocol above provides a means for certifying the operation against fractional amplitude errors thatare shared by the four tones in either of the two pulses.With respect to this error model, whether we check for abright state immediately or delay the flag query, the dissipative transfer of leftover population to the bright stateeither leaves the ion in the bright state or accomplisheserrorless operation of the gate. The overall probability oferror-free operation is (1 sin2 (δπ1 /2))(1 sin2 (δπ2 /2)) 1 2(δπ/2)2 (where δπ is an average error during thissequence) and for uncorrelated errors, the overall errorprobability is 2 larger than the case without the outcoupling for error detection ( 2(δπ/2)2 ). For single,isolated gates, this accomplishes no error correction, butthe error detection can be used as a means to select instances that are trustworthy against this type of error.For instance, the high-quality rotations that are requiredto perform quantum state or process tomography couldbe certified against conflating errors in the state/processwith this type of error introduced by the tomographyprocess. Perhaps more importantly, more trustworthyNISQ-era [1] computational results can be sorted fromthose that are flagged by this process as containing errors,which may prove a useful way to assess the confidence ofa result.The gate certification idea above is also extendable tomulti-qubit gates and other types of errors. Next, weconsider two examples of particularly troublesome errorsources in the trapped ion platform: qubit addressingerrors, and errors in 2-qubit entangling gates.For trapped ions with hyperfine qubits, an addressedsingle-qubit gate can be driven by a focused laser beamwhere the “microwave” signals are actually in opticalbeatnotes that drive stimulated Raman transitions. Ifthe first step of the certified single-qubit gate describedearlier is applied to ion j by one such laser beam, therecan be a non-negligible amount of light that illuminatesneighboring ions and moves a small amount of their qubitpopulations to their auxiliary manifolds. To deal withthis, the optical pumping beam addressed to ion j canbe augmented by a series of optical pumping beams onthe neighboring ions (or further) that are tuned to cleanout those ions’ auxiliary manifolds. This will either flagan addressing error by producing a bright state, or, morelikely, undo any errant transfer from imperfect addressing by the stimulated Raman beam. The same processcan then be applied for the second half of the gate beingrun on ion j, except that now the clean out will haveall optical pumping beams (including j) set to clean outthe auxiliary manifolds. Addressing errors of the opticalpumping beams themselves are still possible in the firsthalf of the gate, but these will also be flagged by the appearance of a bright state. A lack of bright state qubits,therefore, certifies the gate against both Raman beamand optical pumping beam addressing errors – if all ionsare found to be dark, these addressi