## Introduction

Quantum computers are expected to solve certain computational problems of interest more efficiently than classical computers using state-of-the-art classical algorithms. Notable examples include integer factorization1, unsorted database search2, and quantum dynamics simulations3. Multiple quantum computing platforms are under active development today. One of these platforms is the trapped-ion quantum information processor (TIQIP), which has demonstrated 171Yb+ qubit coherence times in excess of 10 minutes4, single-qubit gate fidelity of 99.9999%5, and two-qubit gate fidelity of 99.9%6,7. In addition, a TIQIP may leverage the all-to-all connectivity between ion qubits. The ability to directly apply a two-qubit gate to any pair of qubits provides TIQIPs an important advantage over other QIPs with limited connectivity8.

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 results9. The shortest quantum program known to date, expected to deliver scientifically meaningful discoveries, requires hundreds of thousands of quantum gates10. Therefore, to address quantum computational problems of broad interest, the two-qubit gate design in TIQIPs must be improved. An efficient procedure that simultaneously implements as many two-qubit 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 two-qubit gates on a TIQIP. Using our efficient, arbitrary, simultaneously entangling (EASE) gates, arbitrary ion-qubit 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, general-purpose, programmable TIQIP, hosted at IonQ11. These new gates pave the way for efficient implementations of large-scale quantum algorithms on a TIQIP.

## Results

### Two-qubit gate on a trapped-ion quantum information processor

The native two-qubit gate on our TIQIP is implemented according to the Mølmer–Sørensen protocol12,13,14, which induces an effective XX-Ising 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 N-ion chain stored in an ion trap. The evolution operator $$\hat{U}$$ that describes this operation is15

$$\hat{U}=\exp \left[\mathop{\sum }\nolimits_{m = 1}^{N}({\hat{\beta }}^{(m)}-{\hat{\beta }}^{(m)\dagger }){\hat{\sigma }}_{x}^{(m)}-i{\sum }_{n\ne m}{\chi }^{(m,n)}{\hat{\sigma }}_{x}^{(m)}{\hat{\sigma }}_{x}^{(n)}/4\right]$$
(1)

where $${\hat{\beta }}^{(m)}=i\mathop{\sum }\nolimits_{p = 1}^{N}{\alpha }_{p}^{(m)}(\tau ){\hat{a}}_{p}^{\dagger }$$ (with motional-mode index p, coupling strength $${\alpha }_{p}^{(m)}$$ between ion m and mode p, the pth motional-mode 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 Pauli-x operator on the mth qubit, and χ(m, n) denotes the degree of entanglement between qubits m and n. To obtain a successful single-pair 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 mn. Similarly, to implement EASE gates between freely chosen pairs of qubits with an arbitrary degree of entanglement for every pair, we require that

1. (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

2. (B)

the second operator’s coefficient χ(m, n) either vanishes (if the ion pair mn is not to be entangled) or computes to a pre-specified degree of entanglement (if the pair is to be entangled).

To satisfy conditions (A) and (B), we individually address participating ions with amplitude-modulated (AM) laser pulses11, where the modulation is performed by dividing the gate time τ into Nseg equi-spaced 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 motional-mode frequencies as ωp, condition (A) implies, for all m and p,

$${\alpha }_{p}^{(m)}(\tau ) =-{\eta }_{p}^{(m)}\int_{0}^{\tau }dt\ {\Omega }^{(m)}(t)\cos (\mu t){e}^{i{\omega }_{p}t}=0\\ \mapsto \sum_{k = 1}^{{N}_{{\rm{seg}}}}{\Omega }_{k}^{(m)}\int_{(k-1)\tau /{N}_{{\rm{seg}}}}^{k\tau /{N}_{{\rm{seg}}}}dt\ \cos (\mu t){e}^{i{\omega }_{p}t}=\hat{M}{{\boldsymbol{\Omega }}}^{(m)}=0,$$
(2)

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

$${\chi }^{(m,n)}= \sum_{k = 1}^{{N}_{{\rm{seg}}}}\sum_{l = 1}^{k}{\Omega }_{k}^{(m)}{\Omega }_{l}^{(n)}\int_{(k-1)\tau /{N}_{{\rm{seg}}}}^{k\tau /{N}_{{\rm{seg}}}}d{t}_{2}\int_{(l-1)\tau /{N}_{{\rm{seg}}}}^{\min ({t}_{2},l\tau /{N}_{{\rm{seg}}})}d{t}_{1}\\ \left[-\sum_{p = 1}^{N}4{\eta }_{p}^{(m)}{\eta }_{p}^{(n)}\sin [{\omega }_{p}({t}_{2}-{t}_{1})]\cos (\mu {t}_{1})\cos (\mu {t}_{2})\right]\\ = {({{\boldsymbol{\Omega }}}^{(m)})}^{T}\hat{\boldsymbol{{{D}}}}^{(m,n)}{{\boldsymbol{\Omega }}}^{(n)}\\ = \left\{\begin{array}{cc}{{\theta }^{(m,n)}} & {{\rm{if}}\,m\,{\rm{and}}\,n\,{\text{are}} \, {\text{to}} \,{\text{be}} \, {\text{entangled,}}} \\ 0 \hfill & {\rm{otherwise,}}\end{array}\right.$$
(3)

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

$${\left({{\boldsymbol{\Omega }}}^{(m)}\right)}^{T}\hat{\boldsymbol{{{S}}}}^{(m,n)}{{\boldsymbol{\Omega }}}^{(n)} =\left\{\begin{array}{cc}{{\theta }^{(m,n)}} & {{\rm{if}}\,m\,{\rm{and}}\,n\,{\text{are}} \, {\text{to}} \,{\text{be}} \, {\text{entangled,}}} \\ 0 \hfill & {\rm{otherwise,}}\end{array}\right.$$
(4)

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 NP-hard, as has been pointed out in the literature17,18. 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.

### EASE-gate 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 motional-mode frequencies of the ion chain, the Lamb–Dicke parameters, the detuning frequency, the desired EASE-gate 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 null-space 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 pulse-shape search-space 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 NEASE participating qubits may require as many as NEASE(NEASE − 1)/2 angle parameters θ(m, n) (see Eq. (3)), we require only Nseg = 2N + NEASE − 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 null-space 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 NEASE − 1. In other words, each participating qubit in an EASE gate is subject to at most 2N + NEASE − 1 linear constraints. Because our approach is completely linear, the EASE-gate pulse shapes that exactly implement the desired operation are obtained in polynomial time.

### Implementation

We implement our EASE-gate protocol on a TIQIP hosted at IonQ11, which can load and control small chains of 171Yb+ ion qubits. Each qubit is optically initialized to a pure quantum state and then manipulated by addressing the qubit with pulses from a mode-locked 355-nm pulsed laser. These pulses can be engineered to drive either single-qubit operations by coupling to the internal (spin) degree of freedom of the ion, or two-qubit 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 13-ion 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 analysis-pulse 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 estimation7,11, we are able to get a lower-bound 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 non-addressed 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 non-involved 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 two-qubit entangling gates than previously reported17 on chains of ions at least twice as long as any previously reported results17,18. The fidelities reported here are markedly lower; however, it should be noted that our results are not corrected for state-preparation 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 solid-state QIPs, where each two-qubit interaction has to be hard-wired during the manufacturing process. TIQIPs can load as many qubits as necessary and employ the EASE-gate 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 circuits19,20—useful for solving an integer factoring problem or computing discrete logarithms over Abelian groups1.

• Multi-control Toffoli gates using global XX gates as a special instance of an EASE gate21—useful, e.g., Grover’s unsorted database search algorithm2, applicable for solving certain satisfiability problems9.

• Fan-in or fan-out CNOTs or various roots of NOTs—useful for realizing the quantum Fourier transform21 or the Bernstein–Vazirani algorithm22.

• Disjoint k-local operators—useful for quantum simulation circuits, including both variational quantum eigensolver23 or Hamiltonian-dynamics simulations10, and the Hidden-shift algorithms24.

To highlight the advantages offered by the EASE operation, in Fig. 4 we show a selection of notable algorithms that benefit from our efficient EASE-gate protocol.

Our EASE-gate 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 power-optimal 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 non-linear, approximate methods used in previous studies17,18 that in general return an approximate pulse-shape solution and require an exponential overhead in the number of segments. Our protocol explains why it was possible in previous studies17 that a certain echo-based pulse-shape 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 equi-spaced segment-based 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 EASE-gate implementations, at the cost of additional degrees of freedom; in the segment-based 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 σz) XX(φ)(1 σz) = (1  1) for any entangling angle φ, where σz is the Pauli-z operator. As an example, this implies that a weak XX interaction or crosstalk, for instance induced between an EASE-participating qubit that sees the pulse shape Ω and a non-EASE-participating qubit that sits adjacent to one of the EASE-participating qubits that sees $$\varepsilon {\boldsymbol{\Omega }}^{\prime}$$, due to the spilled-over beam seen by the nonparticipating qubit, can be removed to first order in ε by repeating twice the interleaving of the σz 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 dots26, neutral atoms27,28, or superconducting circuits29,30, also employ pulse-shape 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 architecturally-inducible 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 Multi-Instruction Multi-Data architectures and today’s personal computers typically employ Single-Instruction Multi-Data 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 well-known Amdahl’s law in classical parallel computing31, we may roughly estimate the speed-up in quantum latency to scale inversely proportional to 1 − p + 2pr/N2, where p denotes the proportion of the quantum computational task that benefits from the simultaneous operations, r = TEASE/TSINGLE denotes the latency overhead of an EASE gate with duration TEASE over a single entangling gate with duration TSINGLE, and the factor N2/2 arises from the capability of the EASE gate to implement up to  ≈N2/2 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.