High-Fidelity Single-Shot Toffoli Gate via Quantum Control

Ehsan Zahedinejad,1, ∗ Joydip Ghosh,1, † and Barry C. Sanders1, 2, 3, 4, ‡

1Institute for Quantum Science and Technology, University of Calgary, Alberta, Canada T2N 1N42Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

3Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics,University of Science and Technology of China, Anhui 230026, China

4Shanghai Branch, CAS Center for Excellence and SynergeticInnovation Center in Quantum Information and Quantum Physics,

University of Science and Technology of China, Shanghai 201315, China(Dated: January 21, 2015)

A single-shot Toffoli, or controlled-controlled-NOT, gate is desirable for classical and quantuminformation processing. The Toffoli gate alone is universal for reversible computing and, accompa-nied by the Hadamard gate, forms a universal gate set for quantum computing. The Toffoli gateis also a key ingredient for (non-topological) quantum error correction. Currently Toffoli gates areachieved by decomposing into sequentially implemented single- and two-qubit gates, which requiresmuch longer times and yields lower overall fidelities compared to a single-shot implementation.We develop a quantum-control procedure to construct a single-shot Toffoli gate for three nearest-neighbor-coupled superconducting transmon systems such that the fidelity is 99.9% and is as fastas an entangling two-qubit gate under the same realistic conditions. The gate is achieved by anon-greedy quantum control procedure using our enhanced version of the Differential Evolutionalgorithm.

PACS numbers: 03.67.Lx, 85.25.Cp, 42.50.Ex

Scalable quantum computing [1, 2] requires a set ofhigh-fidelity universal quantum gates with which to con-struct the circuit [2–4]. Quantum information is thenprocessed within a codespace using fault-tolerant quan-tum error correction. Experimental progress towardsa high-fidelity universal set of gates comprising single-and two-qubit operations has been impressive, exceeding99.9% for single-qubit gates and 99% for an entanglingtwo-qubit gate [3], but an outstanding problem is that(non-topological) quantum error correcting codes requirea gate acting on at least three qubits [5, 6], with the Tof-foli gate [7–9] being optimal. The Toffoli gate is also akey component for reversible arithmetic operations, suchas the modular exponentiation, which is a necessary stepin Shor’s factoring algorithm [10].

The quantum Toffoli gate is to effect a three-qubitcontrolled-controlled-NOT (CCNOT) gate, which meansthat the third qubit is flipped (α|0〉+β|1〉 7→ α|1〉+β|0〉)only if the first two qubits are in the |1〉 state and notflipped otherwise. Thus far Toffoli gates are achieved bydecomposing into single- and two-qubit gates with resul-tant fidelities limited to 81% in a post-selected photoniccircuit [11], 71% in an ion-trap system [7], and 68.5% ina superconducting circuit [8]. We here introduce a non-greedy quantum-control approach for directly construct-ing Toffoli gates based on an enhanced version of theDifferential Evolution (DE) algorithm [12, 13]. We showthat our scheme applied to the three nearest-neighbor-coupled superconducting transmon systems should pro-duce a Toffoli gate operating with 99.9% fidelity andoperating as fast as an entangling two-qubit gate un-der the same conditions. As our quantum-control-based

approach [14] to realizing the Toffoli gate does not re-sort to decomposition, a fast Toffoli gate enables error-correction with high fidelity under this scheme. An ad-ditional valuable benefit of realizing CCNOT directly isthat the Hadamard (H) and CCNOT together make auniversal gate set [15] with significant advantages overthe oft-studied H, π/8 gate and CNOT universal set [2].

Superconducting circuits offer a promising medium forrealizing a high-fidelity CCNOT gate based on quantumcontrol of three nearest-neighbor-coupled superconduct-ing artificial atoms [3] as opposed to a non-scalable cir-cuit quantum electrodynamics architecture that employsjust a few artificial atoms coupled to the superconductingcavity [8, 9]. Our approach is to vary the energy levelsfor each of three individual superconducting atoms usingtime-dependent control electronics, which convenientlydo not require additional microwave control [16]. Thecombination of varying the energy levels and the nearest-neighbor interactions between atoms causes populationmixing as well as generates required dynamic phases foreach of the energy states. A similar strategy has re-cently been employed successfully to design two-qubitcontrolled-Z gates, for which optimal pulses are foundvia a greedy algorithm [17]. We, however, have observedthat existing optimization algorithms (including greedyalgorithms) are insufficient to generate an optimal pulsefor high-fidelity Toffoli gates, and, therefore, developeda non-greedy optimization scheme, referred to here asSubspace-Selective Self-Adaptive DE or SuSSADE. Theoptimization algorithm shapes the time-dependent con-trol pulses such that the system evolution at a specifiedtarget time closely approximates the desired Toffoli gate.

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We consider a linear chain of three nearest-neighbor-coupled superconducting artificial atoms, realized astransmons [3] with distinct locations labeled k = 1, 2, 3.The transmons have non-degenerate discrete energy lev-els, labeled {|j〉k}, with j = 0 for the ground state. Theenergies are anharmonically spaced, with this spacing al-lowed to be dependent on the specific transmon. Whereassuperconducting atoms contain many energy levels, wetruncate all energy levels for j > 3 for each transmon asa CCNOT operates on at most three excitations.

Although the Toffoli gate acts on three qubits as perdefinition, our quantum-control procedure operates onthe first four levels of each transmon. The Hamiltonianthat generates Toffoli acts on the 43-dimensional Hilbertspace H ⊗3

4 with energy basis {|j ∈ {0, 1, 2, 3}〉⊗3}. Wefollow the standard practice of specifying transmon en-ergy levels instead as frequencies with these atomic fre-quencies shifted by frequency of a rotating-frame basistransformation: the shifted frequency of the kth trans-mon is ∆k, and the anharmonicity of the jth level ofthe kth transmon is ηjk. Therefore, the energy of the kth

transmon’s jth level at time t is h(j∆k(t)− ηjk).

Nearest-neighbor transmons couple via an XY interac-tion with coupling strength between the kth and (k+1)th

transmons denoted by gk. The three-transmon Hamilto-nian is thus [16]

H(t)

h=

3∑k=1

3∑j=0

(j∆k(t)− ηjk) |j〉k 〈j|k

+

2∑k=1

gk2

(XkXk+1 + YkYk+1) , (1)

for coupling operators

Xk =

3∑j=1

√j |j − 1〉k 〈j|k + hc,

Yk =−3∑

j=1

√−j |j − 1〉k 〈j|k + hc, (2)

which are higher-dimensional generalizations of Paulispin matrices [16, 18], and hc denotes Hermitian con-jugate.

Here we employ Hamiltonian evolution to realize theCCZ gate, which effects α|0〉+β|1〉 7→ α|0〉−β|1〉 on thethird qubit only if the first two qubits are |1〉. The CC-NOT and CCZ operations are equivalent under the localtransformation CCNOT= [1⊗1⊗H] CCZ [1⊗1⊗H](similar to the equivalence between two-qubit CNOT andCZ gates), with H straightforward to implement as afast single-qubit operation [19, 20]. The CCZ gate isachieved by varying ∆k of each superconducting atomover duration Θ with resultant Hamiltonian-generated

time-ordered (T ) evolution operator

U(Θ) = T exp

−i

Θ∫0

H(τ)dτ

. (3)

Whereas our approach enables generating any desir-able pulse shape for ∆k, here we consider two types ofpulses: piecewise-constant and piecewise-error-function.These time-dependent control pulses are constrainedwithin the frequency-range of a superconducting trans-mon system. We employ the less computationally-expensive piecewise-constant function to demonstrate theexistence of an optimal pulse for high-fidelity Toffoli gate.However, such a piecewise-constant pulse is beyond thereach of existing pulse generators due to the infinitesi-mally small switching times between consecutive steps.The control electronics for superconducting systems isonly capable of generating smooth pulses, which moti-vated us to consider a realistic case for which the con-trol parameters are connected together via smooth er-ror functions, as opposed to discontinuous steps [16, 21].We show that the gate fidelity does not depend on thepulse shape, and only depends on the number of controlparameters. Therefore, without any loss of generality,we choose the less computationally expensive piecewise-constant control function to analyze the fidelity of thedesigned gate against other parameters.

The Hamiltonian evolution (3) describes the unitarydynamics of the system in the absence of decoherence.Decoherence is incorporated by treating each atom asa damped harmonic oscillator characterised by ampli-tude and scattering induced phase-damping rates foreach oscillator. The corresponding timescales are relax-ation time T1 and dephasing time T2, analogous to therates employed for two-level systems [2, 22]. We hereassume T := T1 ≡ T2, which is valid for frequency-tunable transmons [16]. These decohering processesmodify the unitary evolution (3) to a completely-positivetrace-preserving map E(Θ), which is decomposable intoan operator sum as discussed in the Supplementary Ma-terial.

A high-fidelity quantum gate is usually designed bydetermining an optimal control pulse for each fre-quency ∆k neglecting open-system effects such as de-coherence. Performance is assessed for the unitaryevolution (3) projected to the computational subspace:UP(Θ) := PU(Θ)P. The standard figure of meritfor performance of UP(Θ) is the “intrinsic fidelity” (fi-delity neglecting decoherence) with respect to the idealgate, in this case CCZ, so the intrinsic fidelity is [13]

F = 18

∣∣∣Tr(

CCZ† UP(Θ))∣∣∣ with F = 1 if UP(Θ) =

CCZ and 0 ≤ F < 1 otherwise. After determining con-trol pulses that maximize F , decoherence is then incor-porated into the calculation to assess the performanceunder open-system conditions [16].

3

In the presence of decoherence, the efficacy of the non-unitary evolution compared to the target gate is quan-tified by the average state fidelity F , which is calcu-lated as follows. For {|ψk〉 ∈ H ⊗3

2 } the set of three-transmon computational basis states, the non-unitary ex-tension of the unitary evolution (3) transforms a purecomputational basis state |ψk〉〈ψk| into a mixed stateρfinalk . As each basis state |ψk〉 〈ψk| remains invari-

ant under an ideal CCZ gate, average state fidelity

F = 18

∑k

√∣∣〈ψk| ρfinalk |ψk〉

∣∣ quantifies the efficacy of

a quantum gate in the presence of intrinsic as well asdecoherence-induced noise for a given optimal pulse.Whereas F ≈ 99.9% is considered to be a threshold fortopological (surface-code) fault-tolerance for single- andtwo-qubit gates [3], our approach achieves this fidelityeven for the three-qubit CCZ gate subject to realisticconstraints of the control pulses. In this work, unlessotherwise stated, the average state fidelity is referred toas fidelity.

The strategy for controlling the evolution (3) is tovary the frequencies so that energy levels approach eachother but then avoid degeneracies, known as avoided levelcrossings. These avoided crossings mix energy-level pop-ulations and dynamical phases together. This avoided-crossing effect enables shaping the evolution toward thefinal time-evolution operator objective, which is obtainedby maximizing F .

Optimal pulse shapes for each ∆k are obtained by dis-cretizing the time duration Θ into N constant intervals ofduration ∆τ := Θ/N , and the control-problem parame-ter space is spanned by the set of variables {∆k(`∆τ); ` =1, . . . , N} for each k. These control-points are then con-nected via step functions or error functions to constructthe pulse shapes as we described earlier. The CCZ opti-mization problem is non-convex with a 3N -dimensionalparameter space corresponding to N parameters for eachof the three frequencies ∆1,2,3. For a fixed ∆τ , therefore,the dimension of the parameter space increases linearlywith the total time duration Θ, which influences whichoptimization methods work and which do not.

We devise a quantum-control procedure that designsan optimal pulse for a Toffoli gate, which operates as fastas a two-qubit gate [16] with a target intrinsic fidelityof 0.9999. We first use the existing optimization algo-rithms, namely quasi-Newton approach, which employsthe Broyden-Fletcher-Goldfarb-Shanno (BFGS) approxi-mation of the Hessian [23–27], simplex methods [28], sev-eral versions of particle swarm optimization [29, 30], anddifferential evolution (DE) algorithms [13, 31]; however,they all failed to reach the target intrinsic fidelity underthe time-constraint of the fast Toffoli gate. Therefore, weconstruct a new optimization algorithm here to realize aToffoli gate that reaches our target.

Of these optimization approaches, DE yielded the bestfidelity but failed to reach the target due to the well

known problems of searching high-dimensional param-eter spaces [32]. This drawback motivated us to en-hance DE for such high-dimensional problems by insteadbreeding over randomly selected low-dimensional sub-spaces, hence our name Subspace-Selective Self-AdaptiveDE (SuSSADE) algorithm (see Supplementary Material).One of our objectives is to demonstrate (numerically)the capability of SuSSADE with respect to finding a so-lution equally successfully regardless of parameter-spacedimension within the regime that is relevant for currentsuperconducting experiments.

To understand our enhancement, we first briefly reviewstandard DE [31]. DE cooperatively evolves a collectionof trial solutions, called chromosomes, towards an opti-mal solution. Chromosomes are labeled by their locationin the parameter space, and optimization is thus a searchfor the best chromosome in this space. Evolution fromone generation (i.e., chromosomes for one iteration step)to the next is achieved by breeding each chromosomewith three other randomly chosen chromosomes from thesame generation. Breeding yields a single daughter chro-mosome, and only the fittest of the original and daughterchromosome survives. This breeding-and-survival proce-dure continues until either a chromosome reaches the req-uisite F or the number of generations reaches a specifiedupper bound.

Whereas standard DE breeds chromosomes randomlyselected from the entire space, our SuSSADE algorithmis much faster due to breeding being restricted someof the time to a subset of chromosomes drawn from alow-dimensional subspace, i.e., some fixed parametersand some variable parameters. Our algorithm randomlyswitches breeding between the subspace and the wholespace according to the value of an input switch parame-ter S ∈ [0, 1] such that a uniformly distributed randomnumber rj ∈ [0, 1] at generation j restricts breeding tothe subspace if rj < S and breeds in the whole spaceotherwise.

In the extreme case of restricting to one-dimensionalsubspaces, chromosomes can breed only if all but oneof the parameters are the same. We refer to this one-dimensional extreme case as 1DSuSSADE. Henceforth weuse only 1DSuSSADE as it works well with S = 0.14 fordesigning the Toffoli gate.

Here we present two types of pulses that achieve thetarget intrinsic fidelity, and we explore how the perfor-mance of piecewise constant pulses vary with respect tothe total gate time, coupling strength and decoherence-induced noise. The success of our quantum control pro-cedure corresponds to a target intrinsic fidelity of 0.9999and a timescale comparable to a two-qubit gate [16].

Figure 1 shows both piecewise-constant as well aspiecewise-error-function pulses that achieve the target in-trinsic fidelity obtained by optimizing all the parameterswithin the experimental constraints. The CCZ gate cor-responding to Figure 1 requires a total gate time of 26 ns

4

0 5 10 15 20 26−2.5

0

2.5

∆

1

∆2

∆3

⌧ (ns)

A�

1,2,

3(G

Hz)

0 5 10 15 20 26−2.5

0

2.5

∆

1

∆2

∆3

⌧ (ns)

�1,

2,3

(GH

z)

B

FIG. 1: (color online) Optimal pulse shapes for the Toffoligate given as frequency detunings ∆i (for the i = 1, 2, 3) ofthe superconducting atoms, corresponding to A) piecewise-constant and B) error-function-based pulse profiles, as a func-tion of time τ with constant step-size time interval ∆τ = 1ns and with F = 0.9999 and g = 30 MHz. The black dots onboth plots show the control parameters used to optimize theshape of the pulses.

20 22 24 26 280.89

0.93

0.95

0.97

0.99

0.9999

g=20MHz

g=30MHz

g=40MHz

g=50MHz

⇥ (ns)

A

F

20 30 40 500.038

0.039

0.04

0.041

0.042

0.043

0.044

1/⇥

(ns-1

)

B

g (MHz)

FIG. 2: (color online) A) Intrinsic fidelity F vs. total gatetime Θ for various coupling strengths g and B) 1/Θ vs g forF = 0.999. The 3, 4, ◦, and � denote the actual numericalcomputations using 1DSuSSADE, and solid lines depict cubic-fit curves.

given a coupling strength of g = 30 MHz [16, 17]. Com-paring the intrinsic fidelities of Figure 1a and Figure 1bshows that the target fidelity does not depend on theshape of the pulse; rather it depends on the number ofcontrol parameters. In what follows, therefore, we con-sider only the piecewise-constant pulse shapes as theseare computationally less expensive to handle and alsodo not compromise the generality of our results. Weshow in Figure 2 how the (maximized) intrinsic fidelitychanges when the parameters g and Θ are varied withina range commensurate with currently available super-conducting circuits [3]. Figure 2a gives the intrinsic fi-delity as a function of total gate time for various cou-pling strengths g. Figure 2b shows that the total gatetime changes linearly with the coupling g, in order toachieve a given fidelity. Finally, we consider the effect ofdecoherence on the approximate CCZ gate obtained by

20 30 40 50 600.9985

0.999

0.9995

1

T (µs)

F

FIG. 3: (color online) The fidelity F is plotted against thecoherence times T . We assume T = T1 = T2, with T1 andT2 the relaxation time and dephasing time of each transmonrespectively. This assumption is valid for tunable transmons.Each 3 denotes an actual numerical result obtained from thedecoherence calculation. The solid line depicts the cubic-fitcurve.

optimal pulses shown in Figure 1a, and compute the fi-delity F . Amplitude-damping and phase-damping rates(T−1

1 and T−12 ) are treated as the dominant forms of de-

coherence. For fast gates with Θ � T , but with deco-herence more significant than intrinsic errors, an order-of-magnitude estimate yields 1 − F ∼ Θ/T [16], whichis consistent with the numerically evaluated plot of Fvs T in Figure 3. We have employed our quantum con-trol procedure to determine the optimal pulse (Figure 1)for a high-fidelity single-shot three-qubit Toffoli gate. Wecomputed a smooth pulse (Figure 1b), for which the con-trol parameters are separated by 1 ns and connectedvia error functions, thereby ensuring that the pulse iscompatible with the power and bandwidth specificationsof standard control electronics. Applying our approachto the three nearest-neighbor-coupled superconductingtransmon systems produces a fast and high-fidelity Tof-foli gate in 26 ns, which matches the timescale for thetwo-qubit avoided-crossing-based CZ gate.

The longer total gate time, with a fixed ∆τ , generates ahigher-dimensional parameter space for the optimizationalgorithm. The monotonically increasing optimized in-trinsic fidelity in Figure 2a with increasing total time thusdemonstrates the capability of our algorithm for a param-eter regime relevant to superconducting experiments, forwhich alternative algorithms fail. The linear relationshipbetween 1/Θ and g demonstrates that faster gate speedrequires higher coupling. This relation is a character-istic signature for avoided-crossing-based gates (as alsoobtained for avoided-crossing-based two-qubit gates [16])assuming the corresponding optimization algorithm is ca-pable of finding the optimal solution regardless of theparameter space dimension.

The effect of decoherence on the performance of theoptimal Toffoli gate has been explored (Figure 3), andwe interpret the corresponding result as follows: when

5

T � Θ, the effect of thermal noise becomes less dom-inant, and the fidelity of a quantum gate is almost en-tirely contributed by the intrinsic fidelity. Here we havebeen able to design a fast and optimal pulse for a Tof-foli gate using our quantum control approach for whichthe intrinsic fidelity is so high (∼ 99.99%), that the fi-delity (F ∼ 99.9%) is significantly contributed by thedecoherence-induced noise, which is also very small (com-pared to previous realizations) for the state-of-the-art su-perconducting atoms with T ∼ 20− 60 µs [3].

In summary, we have devised a powerful quantum con-trol scheme, named SuSSADE, to design a fast and high-fidelity single-shot Toffoli gate for a scalable chain ofnearest-neighbor-coupled three-transmon system. Thetime required for the Toffoli operation is comparablewith the timescale of two-qubit avoided-crossing-basedCZ gate, which is the key advantage our quantum-controlapproach proffers compared to decomposition-based ap-proaches requiring many such two-qubit gates to im-plement a single CCZ operation. Our three-transmonsystem serves as a module for all 1D and 2D quantumcomputing architectures [33], and therefore one can re-alize our scheme in a large-scale multi-qubit architec-ture, if the undesired couplings are turned off. Our ap-proach demonstrates the efficacy of SuSSADE for design-ing quantum gates as well as yielding the concrete exam-ple of a three-qubit gate required for scalable quantum-error-correction.

This research was funded by NSERC, AITF, the Uni-versity of Calgary’s Eyes High Fellowship Program, andChina’s Thousand Talent Program. We also acknowledgethe support from WestGrid (www.westgrid.ca) and Com-pute Canada Calcul Canada. We thank Alexandre Blaisfor helpful discussions. J.G. gratefully acknowledges dis-cussions with Michael Geller.

∗ Electronic address: ezahedin@ucalgary.ca† Electronic address: ghoshj@ucalgary.ca‡ Electronic address: sandersb@ucalgary.ca

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