Abstract
Efficiently entangling pairs of qubits is essential to fully harness the power of quantum computing. Here, we devise an exact protocol that simultaneously entangles arbitrary pairs of qubits on a trappedion quantum computer. The protocol requires classical computational resources polynomial in the system size, and very little overhead in the quantum control compared to a singlepair case. We demonstrate an exponential improvement in both classical and quantum resources over the current state of the art. We implement the protocol on a softwaredefined trappedion quantum computer, where we reconfigure the quantum computer architecture on demand. Our protocol may also be extended to a wide variety of other quantum computing platforms.
Introduction
Quantum computers are expected to solve certain computational problems of interest more efficiently than classical computers using stateoftheart classical algorithms. Notable examples include integer factorization^{1}, unsorted database search^{2}, and quantum dynamics simulations^{3}. Multiple quantum computing platforms are under active development today. One of these platforms is the trappedion quantum information processor (TIQIP), which has demonstrated ^{171}Yb^{+} qubit coherence times in excess of 10 minutes^{4}, singlequbit gate fidelity of 99.9999%^{5}, and twoqubit gate fidelity of 99.9%^{6,7}. In addition, a TIQIP may leverage the alltoall connectivity between ion qubits. The ability to directly apply a twoqubit gate to any pair of qubits provides TIQIPs an important advantage over other QIPs with limited connectivity^{8}.
While the current progress in TIQIP technology is remarkable, better quality quantum gates are needed to run longer quantum programs and still obtain reliable quantum computational results^{9}. The shortest quantum program known to date, expected to deliver scientifically meaningful discoveries, requires hundreds of thousands of quantum gates^{10}. Therefore, to address quantum computational problems of broad interest, the twoqubit gate design in TIQIPs must be improved. An efficient procedure that simultaneously implements as many twoqubit gates as possible with the least amount of resources will thus accelerate the process of harnessing the power of universal, programmable quantum computers.
In this paper, we devise a new protocol that efficiently and simultaneously implements multiple twoqubit gates on a TIQIP. Using our efficient, arbitrary, simultaneously entangling (EASE) gates, arbitrary ionqubit pairs, overlapping or not, can be entangled with programmable degrees of quantum entanglement. We implement EASE gates by modulating the amplitude of laser pulses that address individual ion qubits that comprise our scalable, generalpurpose, programmable TIQIP, hosted at IonQ^{11}. These new gates pave the way for efficient implementations of largescale quantum algorithms on a TIQIP.
Results
Twoqubit gate on a trappedion quantum information processor
The native twoqubit gate on our TIQIP is implemented according to the Mølmer–Sørensen protocol^{12,13,14}, which induces an effective XXIsing interaction between a pair of qubits. The coupling between the computational states of the qubit pair is mediated by the motional modes of the linear Nion chain stored in an ion trap. The evolution operator \(\hat{U}\) that describes this operation is^{15}
where \({\hat{\beta }}^{(m)}=i\mathop{\sum }\nolimits_{p = 1}^{N}{\alpha }_{p}^{(m)}(\tau ){\hat{a}}_{p}^{\dagger }\) (with motionalmode index p, coupling strength \({\alpha }_{p}^{(m)}\) between ion m and mode p, the pth motionalmode creation operator \({\hat{a}}_{p}^{\dagger }\)—see Fig. 1—and the gate duration τ) denotes the coupling between the computational state of qubit m and the motional modes, \({\hat{\sigma }}_{x}^{(m)}\) is the Paulix operator on the mth qubit, and χ^{(m, n)} denotes the degree of entanglement between qubits m and n. To obtain a successful singlepair XX gate, we require that the first term in Eq. (1) and all χ^{(m, n)} vanish, except for χ^{(m, n)} of the targeted ion pair m, n. Similarly, to implement EASE gates between freely chosen pairs of qubits with an arbitrary degree of entanglement for every pair, we require that
 (A)
the first operator \({\hat{\beta }}^{(m)}\), which represents the coupling between motional modes of the ion chain and the computational states of the qubits, vanishes at the end of the evolution, and that
 (B)
the second operator’s coefficient χ^{(m, n)} either vanishes (if the ion pair m, n is not to be entangled) or computes to a prespecified degree of entanglement (if the pair is to be entangled).
To satisfy conditions (A) and (B), we individually address participating ions with amplitudemodulated (AM) laser pulses^{11}, where the modulation is performed by dividing the gate time τ into N_{seg} equispaced segments and allowing the amplitude to vary from one segment to the next.
Denoting the amplitude of the pulse Ω^{(m)}(t) applied to ion m during segment k as \({\Omega }_{k}^{(m)}\), the laser detuning from the carrier frequency as μ and the motionalmode frequencies as ω_{p}, condition (A) implies, for all m and p,
where \({\eta }_{p}^{(m)}\) denotes the coupling constant (Lamb–Dicke parameter) for qubit m and mode p (see also Fig. 1), \({\hat{\boldsymbol{{M}}}}\) is the matrix with elements that are the segmented integrals shown above, and Ω^{(m)} is the vector of \({\Omega }_{k}^{(m)}\). Likewise, in the segmented form, condition (B) implies
where \({\hat{\boldsymbol{{{D}}}}}^{(m,n)}={\hat{D}}^{(n,m)}\) is the triangular matrix with elements that are the segmented double integrals and the angle parameters \({\theta}^{(m, n)}\) denote the desired degree of entanglement between the qubit pair (m, n). We note that, according to Eq. (1), the desired evolution to be induced between qubits m and n is \(\exp [i({\chi }^{(m,n)}+{\chi }^{(n,m)}){\sigma }_{x}^{(m)}{\sigma }_{x}^{(n)}/4]\). Since the χs are scalars, \({\chi }^{(m,n)}+{\chi }^{(n,m)}={\chi }^{(m,n)}+{({\chi }^{(m,n)})}^{T}\). Therefore, the constraint Eq. (3) may be rewritten as
where \({\hat{\boldsymbol{{{S}}}}}^{(m,n)}=[{\hat{D}}^{(m,n)}+{({\hat{D}}^{(m,n)})}^{T}]/2\) is a symmetric matrix. The problem of finding the amplitude vectors Ω satisfying the two conditions Eq. (2) and Eq. (4) can, in principle, be written in the form of a quadratically constrained quadratic program (QCQP)^{16}, which is in general NPhard, as has been pointed out in the literature^{17,18}. However, our problem is fully specified by the two equations, Eqs. (2) and (4), which is a special case of QCQP. The vectors Ω that satisfy Eq. (2) and Eq. (4) can be solved exactly in polynomial time using a linear approach.
EASEgate protocol
Figure 2 shows a flowchart that outlines our linear approach to produce pulse shapes that implement an EASE gate. Once the experimental parameters, such as the number and positions of the ion qubits, the motionalmode frequencies of the ion chain, the Lamb–Dicke parameters, the detuning frequency, the desired EASEgate duration, the number of AM segments, and the qubit pairs with corresponding degrees of entanglement are specified, our protocol constructs the \(\hat{\boldsymbol{{M}}}\)matrix in Eq. (2). The nullspace vectors of \(\hat{\boldsymbol{{M}}}\) are then computed. They span a vector space from which we draw pulse shapes that satisfy Eq. (4).
To find a suitable pulse shape that requires minimal laser power, an important experimental concern, the \(\hat{\boldsymbol{{S}}}\) matrix in Eq. (4) is first projected onto the null space of \(\hat{\boldsymbol{{M}}}\). The eigenvector c with the largest absolute eigenvalue of the projected matrix is then guaranteed to require the minimal power possible, measured according to the sum of squares of the individual amplitudes \({\Omega }_{l}^{(m)}\). This methodology can then be iterated to find the pulse shapes for all ion qubits involved in the EASE gate (see Supplementary Notes 1 and 2 for theoretical details) by considering the pulseshape searchspace for a given qubit to be the intersection between the full null space and a subspace orthogonal to the space of previously identified pulse shapes for ions that the given qubit needs to be decoupled from.
We note that, even though an EASE gate with N_{EASE} participating qubits may require as many as N_{EASE}(N_{EASE} − 1)/2 angle parameters θ^{(m, n)} (see Eq. (3)), we require only N_{seg} = 2N + N_{EASE} − 1 as the minimal number of segments, which is sufficient to satisfy all χ^{(m, n)} relations and \({\alpha }_{p}^{(m)}\) conditions. This is enabled by the fact that, for every additional participating qubit, we may start with the full set of nullspace vectors that always satisfy condition (A), and the number of relations with respect to each of the participating qubits, according to condition (B), is at maximum N_{EASE} − 1. In other words, each participating qubit in an EASE gate is subject to at most 2N + N_{EASE} − 1 linear constraints. Because our approach is completely linear, the EASEgate pulse shapes that exactly implement the desired operation are obtained in polynomial time.
Implementation
We implement our EASEgate protocol on a TIQIP hosted at IonQ^{11}, which can load and control small chains of ^{171}Yb^{+} ion qubits. Each qubit is optically initialized to a pure quantum state and then manipulated by addressing the qubit with pulses from a modelocked 355nm pulsed laser. These pulses can be engineered to drive either singlequbit operations by coupling to the internal (spin) degree of freedom of the ion, or twoqubit operations by coupling to both the internal and external (collective motional) degrees of freedom. We realize EASE gates by coupling the internal and external degrees of freedom of many ions simultaneously with segmented AM laser pulses.
In particular, we implemented EASE gates to fully entangle qubits in multiple disjoint pairs in a system with 11 ion qubits on a 13ion chain. Of these qubits, up to 5 pairs (10 qubits) were simultaneously entangled. We then performed partial output state tomography on each entangled state by measuring the parity of the entangled pairs as a function of an analysispulse angle (shown in Fig. 3), and also measuring the even parity population without applying analysis pulses. By extracting the amplitude of the measured parity and populations via maximum likelihood estimation^{7,11}, we are able to get a lowerbound estimate of the fidelity of the performed EASE gate. For our implementation with five simultaneous gates (Fig. 3a), we estimate an average gate fidelity of \(F=88.{3}_{1.0}^{+1.6} \%\). For the case in which we applied five gates sequentially (Fig. 3b), we estimate an average gate fidelity of \(F=92.{0}_{1.4}^{+0.8} \%\). The given errors on fidelity represent a 1σ confidence interval on the maximum likelihood estimation used to determine the fidelity.
We use the same technique to estimate any residual entanglement with nonaddressed ions, due predominantly to optical crosstalk, by determining the overlap of any pair with the fully entangled Bell state we are trying to prepare. For pairs with one ion participating in a gate, the fidelity is ideally F = 25%, which corresponds to a fully mixed state. For pairs where neither ion participates in an applied gate, we expect to have F = 50% because the initial pure state has 50% overlap with the Bell state we are trying to prepare. The 50 noninvolved pairs have \(\delta F=2.{3}_{1.6}^{+2.1} \%\) average deviation from the ideal fidelity for the five simultaneously applied gates (Fig. 3a). In the case of five sequentially applied gates (Fig. 3b), we see an average deviation from the ideal fidelity of \(\delta F=0.{9}_{1.0}^{+2.4} \%\). In these results, we have performed more simultaneous twoqubit entangling gates than previously reported^{17} on chains of ions at least twice as long as any previously reported results^{17,18}. The fidelities reported here are markedly lower; however, it should be noted that our results are not corrected for statepreparation and measurement errors.
Discussion
Because a TIQIP can induce couplings between arbitrary pairs of qubits by simply switching on and off pairwise interactions, the EASE gates developed and demonstrated here can readily be implemented on a TIQIP through software alone. This is in contrast to other quantum hardware platforms such as a solidstate QIPs, where each twoqubit interaction has to be hardwired during the manufacturing process. TIQIPs can load as many qubits as necessary and employ the EASEgate protocol to simultaneously implement any combinations of simultaneously addressible Ising interactions with little to no extra cost at the hardware level.
A host of quantum algorithms benefit from the ability to implement EASE gates. These algorithms tend to contain an orderly structure such that the circuit may be manipulated to reveal multiple Ising interactions applied simultaneously. For instance:
Quantum arithmetic circuits^{19,20}—useful for solving an integer factoring problem or computing discrete logarithms over Abelian groups^{1}.
Multicontrol Toffoli gates using global XX gates as a special instance of an EASE gate^{21}—useful, e.g., Grover’s unsorted database search algorithm^{2}, applicable for solving certain satisfiability problems^{9}.
Fanin or fanout CNOTs or various roots of NOTs—useful for realizing the quantum Fourier transform^{21} or the Bernstein–Vazirani algorithm^{22}.
Disjoint klocal operators—useful for quantum simulation circuits, including both variational quantum eigensolver^{23} or Hamiltoniandynamics simulations^{10}, and the Hiddenshift algorithms^{24}.
To highlight the advantages offered by the EASE operation, in Fig. 4 we show a selection of notable algorithms that benefit from our efficient EASEgate protocol.
Our EASEgate protocol is linear and the pulse shapes we obtain exactly solve the problem and induce the desired quantum operation with up to N(N − 1)/2 angle parameters θ^{(m, n)} with minimal control overhead, i.e., linear in N, comparable to a single XX gate in terms of the number of segments. The shapes are generated in time polynomial in the system size and are poweroptimal for the AM approach when used for a single XX gate with a fixed number of segments, since, in this case, the EASE protocol produces the pulse vector Ω with the minimal possible norm that implements a single entangling gate. This is in contrast to the nonlinear, approximate methods used in previous studies^{17,18} that in general return an approximate pulseshape solution and require an exponential overhead in the number of segments. Our protocol explains why it was possible in previous studies^{17} that a certain echobased pulseshape ansatz worked well for applying simultaneous gates on disjoint pairs of qubits—the shape automatically satisfies the entanglement requirement condition (B) and the infidelity owing to the imperfect decoupling from the motional modes, due to condition (A), may be minimized by navigating through the null space of \(\hat{\boldsymbol{{M}}}\). Furthermore, our protocol enables us to entangle pairs of qubits with overlapping qubits.
Our protocol is scalable and is guaranteed to work for any modulation that admits a linear construction, such as the equispaced segmentbased AM approach explored here or a more general approach demonstrated in ref. ^{25} (see Sec. S12 therein). In addition, because, again, our protocol admits a linear construction, we can readily take advantage of the high degree of stabilization with respect to external parameter fluctuations demonstrated in ref. ^{25} directly in the EASEgate implementations, at the cost of additional degrees of freedom; in the segmentbased AM method, this translates to an additional number of segments. The improved stability will likely lead to a better gate fidelity. Furthermore, we could leverage the first circuit identity that appears in Sec. IV of ref. ^{21} to remove certain crosstalk errors to first order: Note that the circuit identity reads XX(φ)(1 ⊗ σ_{z}) XX(φ)(1 ⊗ σ_{z}) = (1 ⊗ 1) for any entangling angle φ, where σ_{z} is the Pauliz operator. As an example, this implies that a weak XX interaction or crosstalk, for instance induced between an EASEparticipating qubit that sees the pulse shape Ω and a nonEASEparticipating qubit that sits adjacent to one of the EASEparticipating qubits that sees \(\varepsilon {\boldsymbol{\Omega }}^{\prime}\), due to the spilledover beam seen by the nonparticipating qubit, can be removed to first order in ε by repeating twice the interleaving of the σ_{z} gates on, e.g., a nonparticipating qubit and the EASE gate with half the desired entanglement angles. This costs, in our approach, a factor two in the number of segments used to implement an EASE gate. Our approach can be further generalized to mitigate higher order crosstalk errors at the cost of more segments.
We note that other quantum information processor architectures, such as those based on quantum dots^{26}, neutral atoms^{27,28}, or superconducting circuits^{29,30}, also employ pulseshape techniques to induce desired quantum operations. While the evolution operators for these approaches are not identical to the one considered here, the motivation behind the pulse shaping is the same: Remove the unwanted coupling while preserving the desired interaction from the architecturallyinducible Hamiltonian. We anticipate that the kind of efficient, linear approach we show here may be applicable for other qubit technologies with further research.
Classical supercomputers employ MultiInstruction MultiData architectures and today’s personal computers typically employ SingleInstruction MultiData architectures. These parallel architectures have contributed significantly to sustaining the growth of classical processing power in the era where the frequency scaling of the processors has halted. Likewise, we expect the EASE protocol we explore in this paper to significantly boost the power of quantum computing, unlocking its ability to implement multiple entangling gates efficiently. Akin to the wellknown Amdahl’s law in classical parallel computing^{31}, we may roughly estimate the speedup in quantum latency to scale inversely proportional to 1 − p + 2pr/N^{2}, where p denotes the proportion of the quantum computational task that benefits from the simultaneous operations, r = T_{EASE}/T_{SINGLE} denotes the latency overhead of an EASE gate with duration T_{EASE} over a single entangling gate with duration T_{SINGLE}, and the factor N^{2}/2 arises from the capability of the EASE gate to implement up to ≈N^{2}/2 entangling gates at a time. We believe simultaneously entangling gates, such as the EASE gates developed in this paper, will help ensure continued growth of the power of quantum processors, even when we encounter resource limitations per qubit.
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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Acknowledgements
The authors would like to thank all members of the IonQ, Inc. team, specifically the efforts of Coleman Collins, Alex Milstead, Jonathan Mizrahi, Kai Hudek, and Jamie David WongCampos for design and implementation of experimental system components, as well as assistance with interpretation and visual illustrations of figures presented here.
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R.B., Y.N., and N.G. devised the linear nullspace method for the efficient construction of pulses. N.G. devised the EASE protocol under Y.N.’s supervision. N.G., M.L., R.B., and Y.N. carried out the in silico implementation of the protocol. The apparatus was designed and built by K.B., K.W., V.C., J.A., S.D., N.P., and JS.C. and experimental data was collected and analyzed by K.B. and K.W.; Y.N., K.W., R.B., N.G., and K.B. prepared the paper with input from all authors.
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Grzesiak, N., Blümel, R., Wright, K. et al. Efficient arbitrary simultaneously entangling gates on a trappedion quantum computer. Nat Commun 11, 2963 (2020). https://doi.org/10.1038/s41467020167909
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