Noise Analysis for High-Fidelity Quantum Entangling Gates in an Anharmonic Linear Paul Trap
The realization of high fidelity quantum gates in a multi-qubit system, with a typical target set at , is a critical requirement for the implementation of fault-tolerant quantum computation. To reach this level of fidelity, one needs to carefully analyze the noises and imperfections in the experimental system and optimize the gate operations to mitigate their effects. Here, we consider one of the leading experimental systems for the fault-tolerant quantum computation, ions in an anharmonic linear Paul trap, and optimize entangling quantum gates using segmented laser pulses with the assistance of all the collective transverse phonon modes of the ion crystal. We present detailed analyses of the effects of various kinds of intrinsic experimental noises as well as errors from imperfect experimental controls. Through explicit calculations, we find the requirements on these relevant noise levels and control precisions to achieve the targeted high fidelity of for the entangling quantum gates in a multi-ion crystal.
The trapped ion system is one of the most promising candidates for large-scale quantum computing because of its long coherence time, nearly perfect initialization and detection methods, and also the strong laser-mediated Coulomb interaction among ions which facilitates long-range entangling gates Cirac and Zoller (1995); Blatt and Wineland (2008); Monroe and Kim (2013). Many pioneering works have laid the building block of a scalable ion-trap quantum computer Blatt and Wineland (2008); Monroe and Kim (2013); Ladd et al. (2010). For example, recently, fidelity higher than 99.99% for a single-qubit gate and 99.9% for an entangling gate in a two-ion crystal have been reported Ballance et al. (2016); Gaebler et al. (2016). Meanwhile, some important quantum algorithms, such as Shor’s algorithm and quantum error correction, have been demonstrated in small scale Monz et al. (2016); Chiaverini et al. (2004); Schindler et al. (2011).
One remaining problem is how to scale up the system. For a small number of ions, one scheme to realize the entangling gate, known as the Molmer-Sorensen (MS) gate, has been proposed for almost two decades Sørensen and Mølmer (1999). It utilizes a single phonon mode of the ion crystal, typically the center-of-mass mode, to mediate a coupling between two ions’ internal states, which is insensitive to the phonon number. However, as the number of ions increases, the motion of the ion crystal becomes progressively more complex and the crosstalk among different collective modes can lead to errors in the quantum gate Blatt and Wineland (2008). A straightforward approach to suppress this crosstalk is to weaken the laser driving, but at the cost of increasing the gate time with the number of ions. One possible solution is to use an architecture called the quantum charge-coupled device Kielpinski et al. (2002); Hensinger et al. (2006); Monroe and Kim (2013), where entanglements are first generated in individual zones and are then distributed to other regions by a classical ion shuttling technique. Such a shuttling, however, demands exquisite control of ion positions. In this paper we will focus on a different approach, where all the collective modes are utilized to perform optimized entangling gates Shi-Liang Zhu et al. (2006). In this way, the existence of multiple phonon modes is no longer a source of error. One can then use segmented laser pulses to optimize the gate performance.
Many theoretical and experimental works have been done along this path Zhu et al. (2006); Lin et al. (2009); Choi et al. (2014); Wang et al. (2015). For example, it was proposed that a suitable quartic potential can be applied to make the ion spacing of a linear chain more uniform Lin et al. (2009). In this setup, the ion structure will be more stable against the zigzag shape and it has a narrower transverse phonon spectrum, allowing more efficient cooling and control.
The original scheme of Ref. Shi-Liang Zhu et al. (2006) uses many approximations. However, a detailed and systematic error analysis, which is essential for high gate fidelity above the fault-tolerant error threshold, is still lacking. On the other hand, such an error analysis has been made in Ref. Ballance (2017) for a two-ion crystal, but its scheme and the numerical methods cannot be directly applied to a multi-ion crystal. In this paper, we thoroughly examine the approximations made in the scalable scheme and the influence of fluctuation in gate parameters. Many of our analyses can also be applied to other protocols based on extensions of the MS gate. Here we focus on a one-dimensional (1D) ion crystal, while a generalization to other structures is straightforward.
The paper is organized in the following way. In Sec. II we review the scheme to realize the XX entangling gate between any pair of ions in a large ion crystal. Then in Sec. III an example is presented for a chain of 19 ions. We choose this size of the ion crystal since it is the size of the current experimental platform for the demonstration of a logic qubit. The formalism and many of the analyses in this paper are directly applicable to ion crystals of any size. We numerically optimize the gate parameters to realize high-fidelity entangling gates between ions with different separations. Stability of the gate under fluctuation in these parameters is then discussed, together with a comprehensive list of the source of errors from the approximations and the neglected physical effects. We then conclude in Sec. IV. Appendix A describes the technique to find the ions’ equilibrium configuration and the collective phonon modes. Appendix B discusses in depth the effects of higher order terms neglected in our formulation. Then we examine a key consideration in our scheme, the asymmetry in laser beams, in Appendix C. Finally in Appendix D we discuss how the errors from imperfect gate design accumulate when the gates are applied repeatedly.
Ii Entangling Gates in a Linear Paul Trap
ii.1 Hamiltonian and Time Evolution Operator
Consider ions in a linear Paul trap along the axis. A suitable quartic potential can be applied in the direction through external electrodes, making the spacings of ions nearly uniform. For typical experimental parameters, the micro-motion is small and can be neglected. Then we can calculate the equilibrium configuration as well as the collective oscillation modes. See Appendix A for more details about the derivation.
We start from a three-level approximation of the ion’s level structure (Fig. 1). Later we will adiabatically eliminate the excited state to attain the two-level approximation. The free Hamiltonian of this system is
where is the energy difference between and (typically two hyperfine “clock” states of the ion Choi et al. (2014)) and is the energy splitting between and an excited state . is the frequency of the -th phonon mode, with the corresponding annihilation (creation) operator ().
For simplicity, let us first consider one ion, say ion , in the chain of ions. Suppose two beams of laser (with frequencies and wave vectors ) are shined on the ion to off-resonantly couple states and , and and , with Rabi frequencies and respectively (see Fig. 1). This corresponds to the following coupling Hamiltonian
where and are chosen to be real. The time dependence of the Rabi frequency has been omitted for convenience. Define as the single-photon detuning and as the two-photon detuning. Here we assume so that we can neglect other two-photon processes between and .
Now we perform a unitary transformation characterized by with
Then the Hamiltonian in the transformed frame, a.k.a. the Hamiltonian in the interaction picture, is given by
where is the position operator of ion at time , under the free evolution of the collective phonon modes. Here we have made the rotating wave approximation (RWA) with the requirement .
Assume so that the excited state can be adiabatically eliminated, where is the spontaneous emission rate of the excited state. The effective coupling between the state and is given by
where , . The state and are coupled by an effective Rabi frequency . Later, for simplicity, we drop the superscript and denote the effective Rabi frequency on ion by .
The laser also produces AC Stark shift on the two levels. By suitably choosing the relative intensity of the two laser beams and the detuning with respect to the excited state, we can make the shifts on the two levels nearly the same Wineland et al. (2003); Campbell et al. (2010). We will discuss more about this effect in Sec. III.
This effective coupling depends on the relative phase of the two laser beams and therefore the fluctuation on their paths. This problem can be solved by adding a third laser beam to form two pairs of Raman transitions, with detuning and wave vector difference along the direction (see Fig. 2). This is known as the phase-insensitive geometry Lee et al. (2005). We will also briefly discuss the relevance to the phase-sensitive geometry at the end of this subsection.
The effective Rabi frequencies of both pairs are chosen to be . Here we assume , so that is nearly the same for both pairs, with a relative error of the order . Suppose the initial phase differences for the two pairs are and . Then the total interaction Hamiltonian can be written as
where and are called the motional phase and the spin phase Lee et al. (2005). The subscript is used to show that these phases pertain to ion . Small fluctuation in beams’ paths causes opposite changes in and , so is robust against fluctuation. On the other hand, does change, but it can be quite stable during one gate time. As we will show later, the gate fidelity is not sensitive to a constant so long as the phase is the same for both ions. Finally we will choose and , but for the moment let us keep them in the formulae for completeness.
We further define to simplify and drop the superscript on :
The transverse position of ion can be quantized as
where () characterizes the -th normalized mode vector of the collective oscillation. The summation over is limited to the transverse modes along the direction. This can be done because the small oscillations along directions are separable (see Appendix A for more details).
With the Lamb-Dicke parameter , we get
We can expand this expression according to the power of :
The zeroth order term is a single-qubit operation and commutes with other terms. So we can drop it now and apply a single-qubit rotation after the entangling gate to compensate its effect. Actually for the examples considered in Sec. III, we will show that such a compensation is unnecessary. Here we keep terms up to the second order, but we will show later that the error in the fidelity is of the order .
When the lasers are shined on two ions, we get the interaction-picture Hamiltonian
where the summation of is over the two ions and
Unitary evolution in the interaction picture is obtained by the Magnus expansion
describe the coupling between the spin and phonon modes, and
is the coupling between the two spins. Roughly speaking, the terms are displacement operations on the phonon modes conditioned on the spin state of each ion, and the terms are single-spin rotations conditioned on the phonon numbers of each mode. We need to suppress these terms while maintain a large spin-spin coupling to realize the entangling gate. Here again we keep terms up to the second order in and retain only diagonal terms in [Eq. (17)]. An error analysis is performed in Sec. III.2 and Appendix B. In the above derivation we have also dropped a global phase, which has no effect on the entangling gate.
If the effective Rabi frequencies of the laser beams on the two ions are always proportional, e.g. when the lasers come from a single beam through a beam splitter, the expression of can be simplified as
In this way we recover Eq. (2) of Ref. Shi-Liang Zhu et al. (2006).
In the above derivation we assumed a phase-insensitive laser configuration. It is also possible to use the phase-sensitive geometry for the entangling gate with the spin phase being cancelled by a Ramsey-like gate design Lee et al. (2005); Gaebler et al. (2016). However, due to the difference in the resulting Hamiltonian and hence the different commutation relation, a similar expansion in the Lamb-Dicke parameter leads to infinitely more terms. It seems to us that there is no easy justification to throw away these terms for the model we are considering, so we will not go into further details here. Nevertheless, except for the higher order terms in Lamb-Dicke parameters, our other analyses in Sec. III can still be applied to the phase-sensitive setup.
ii.2 XX Entangling Gate and Fidelity
If and for all the modes and both of the ions, for both ions, and for the ions of interest, and , the time evolution operator will be an ideal XX entangling gate. In the basis of , , , where , we have
The subscript and the dependence on have been dropped.
If the initial internal state is and the vibrational modes are in the thermal state with a temperature , the ideal final state is , while the actual state we get is , where means the partial trace over all the motional modes. Then we can use the fidelity to characterize the similarity between these two states and therefore between the ideal and the real gates.
However, the above method depends on the initial state . For a state-independent measure of the similarity between and , we can use the average gate fidelity Nielsen (2002)
where the integration is over the Fubini-Study measure Bengtsson and Zyczkowski (2006). For the moment we assume the spin phases for the two ions, i.e. , . Later we will discuss the effects of nonzero spin phases in Sec. III.1.
Let us express in the above basis:
where , , , are the phases gained by the , , , states, respectively. Note that they are actually operators in the subspace of phonon modes.
Accurate up to second order diagonal terms in , we have
where is the displacement operator of the -th mode.
For an arbitrary operator (not necessarily Hermitian)
lengthy but straightforward calculation shows that
We have used the following formulae in the derivation:
The average gate fidelity can then be written as Nielsen (2002)
where and is an orthogonal basis of unitary operators such that . Here we can choose . Using Eq. (29) we finally obtain
terms [Eq. (18)] appear quadratically in the fidelity, hence its contribution is and is neglected. If and the average phonon number for a typical mode is , the error from neglecting higher order terms is of the order . The fact that there are independent transverse modes has already been included because the coefficient for each mode is also modulated by the vectors, which are normalized to 1.
Suppose the laser intensities on the two ions are always proportional and that their phases are locked such that , then we get . The above expression can be simplified as
This average gate fidelity is slightly higher than Eq. (3) of Ref. Shi-Liang Zhu et al. (2006), where a special initial state is used.
Also notice that if , the gate is close to another ideal entangling gate , which is different from only by local operations. In this case the gate fidelity can be calculated in a similar way and the final result is
From now on, by fidelity we mean the average gate fidelity if not specifically mentioned. We will drop the overline on for convenience.
In the experiment, we can set the laser beams on the two ions to be the same. We can divide the laser sequence into equal segments and in each segment let the Rabi frequency be a constant. Define a real column vector corresponding to the Rabi frequency of each segment, and we get
where is a row vector whose -th component is
and is an by matrix whose component is
We can further define a symmetric matrix such that . By suitably scaling , we can always set . Then in the limit of small (high fidelity), the fidelity can be approximated as
By definition, is a Hermitian matrix, but actually we can express it in a real symmetric form:
Now we want to minimize under the constraint . For this purpose, we use the method of Lagrange multiplier and consider the optimization of :
We only need to solve this generalized eigenvalue problem and find the eigenvalue with the smallest absolute value. The corresponding eigenvector, with suitable normalization, gives us the optimal . (See also Appendix A of Ref. Lin (2010).)
We remark that for realistic experimental parameters, the effective Rabi frequency cannot be too large. This means that the above optimization should be performed under another inequality constraint. This problem is generally hard to solve, so instead we use the method mentioned above and then discard solutions with unrealistic .
Iii Systematic Errors and Experimental Noise
The gate fidelity realized in the experiment is always less than 1. This is due to the approximations in the formulation, imperfections in the gate design, as well as noise and errors in the experiment. In this section we analyze these sources of errors in detail.
In order to estimate the influence of each error term, we consider a specific example of mapping a 17-qubit surface code for quantum error correction into a linear chain of ions Tomita and Svore (2014); Trout et al. (2018) (see Fig. 3 for the mapping).
For this purpose, diamond norm may be a better measure of the gate performance, but we focus on average gate fidelity here as it is easier to treat theoretically. We will discuss their difference in Sec. III.4. For realistic parameters, we choose MHz, and consider a chain of 19 ions with the two ions at the ends only used for cooling. An anharmonic potential is applied along the axis, which is specified by m and (see Appendix A for the definition). In this way the central 17 ions will have a nearly uniform spacing with an average of m and a relative standard deviation of 2.3%.
Under these conditions, the spectrum of the transverse normal modes is very narrow (within 0.9% of ). Hence it is possible to use sideband cooling method to cool the transverse motion down to about 0.5 phonon per mode or less. Doppler cooling can also be used if the trapping can be stronger. For counter-propagating laser beams along the directions with nm Campbell et al. (2010), we have a detuning THz and . (Actually there are two excited states with a fine-structure splitting of THz, and the laser detuning is specially chosen to minimize the differential AC Stark shift. We will come back to this point when discussing about the AC Stark shift; but otherwise we will just use one value of to estimate the order of magnitude for the other error terms.) The Lamb-Dicke parameter is then for all the transverse modes.
iii.1 Optimized Gate Design and Sensitivity to Tunable Parameters
In order to perform the stabilizer measurement in the surface code, we need to achieve two-qubit entangling gates between nearest neighbor qubits in Fig. 3, that is, ion pairs with one, three and five ion separations in the linear chain. To find the optimal parameters for a high-fidelity gate, we use Eqs. (44) and (46) to estimate the gate fidelity and to solve the optimal pulse sequence. We then scan the gate time , detuning and number of segments to find a combination with the desired fidelity.
For example, Fig. 4 shows the gate infidelity () for the entangling gate between ion 1 and ion 4 as a function of detuning for a fixed gate time s and three possible segment numbers .
As we can see, increasing the number of segments generally reduces the gate infidelity. We also notice that there are multiple local minima in the gate infidelity. Therefore, we do not attempt to find the “best” solution, but rather look for solutions that are “good enough”. That is, the solution needs to achieve high gate fidelity in the ideal case, and it should also be robust against errors in these control parameters, which may arise from imperfect calibration, finite resolution or random fluctuation in the experiment. Specifically, we perturb the gate parameters at local minima of plots similar to Fig. 4 when scanning these parameters and keep the ones that are most insensitive to the noise. We will assume that these noises are “slow” such that they stay constant during one gate period. This assumption is reasonable because typically the high-frequency noise will be weak in the experiment. For instance, Ref. Ballance (2017) considers the influence of high-frequency noise in a two-ion crystal and the experimental noise level is found to be about one order of magnitude lower than what is allowed for an error of . Also note that the same technique to optimize the gate design has been applied in Ref. Trout et al. (2018), but the number of segments and the gate time we use here are generally larger because of this additional requirement of robustness.
Below we show the results for ion pairs with three typical separations. For experimentally achievable effective Rabi frequencies, we only present solutions satisfying MHz at all times.
Ion 5 and ion 6 (separation 1):
We use segments and s. Laser sequence is optimized for (Fig. 5). For the sensitivity to control parameters, in Fig. 6 we show how the gate infidelity changes under a shift in detuning of kHz, in the global laser intensity of , in gate time of s, as well as the effect of a nonzero . (See Eq. (6) for the definition. Here the motional phase is assumed to be equal for both ions.) For parameters fluctuating inside these ranges, the gate infidelity is always below .
Ion 1 and ion 4 (separation 3):
We use segments and s. Laser sequence is optimized for (Fig. 7), but then for the robustness under fluctuation in detuning (where positive and negative shifts have asymmetric effect), the gate is performed at the detuning kHz with a slight rescaling of the laser intensity. (See Appendix D for more details about this rescaling, which aims to reduce the accumulation of errors when multiple gates are applied.) Therefore in Fig. 8 the smallest infidelity does not always appear at the center of the parameter range.
Ion 9 and ion 14 (separation 5):
As we can see in Figs. 6, 8 and 10, a nonzero but constant does not influence the fidelity significantly. This justifies the use of the phase-insensitive setup, which suppresses the fluctuation in but allows to change over different gates. Nevertheless, we still need to set initially for the desired XX entangling gate: by taking in Eq. (37) with small spin phases and , we can shown that it causes an infidelity . Imbalance between and should also be small: numerically we find that the infidelity scales as , thus we need