Scalability of quantum computation with addressable optical lattices

Scalability of quantum computation with addressable optical lattices

T. R. Beals Department of Physics, University of California, Berkeley, California 94720, USA    J. Vala Current address: Department of of Mathematical Physics, National University of Ireland, Maynooth, Co. Kildare, Ireland Department of Chemistry and Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA    K. B. Whaley whaley@berkeley.edu Department of Chemistry and Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA
July 16, 2019
Abstract

We make a detailed analysis of error mechanisms, gate fidelity, and scalability of proposals for quantum computation with neutral atoms in addressable (large lattice constant) optical lattices. We have identified possible limits to the size of quantum computations, arising in 3D optical lattices from current limitations on the ability to perform single qubit gates in parallel and in 2D lattices from constraints on laser power. Our results suggest that 3D arrays as large as sites (i.e., qubits) may be achievable, provided two-qubit gates can be performed with sufficiently high precision and degree of parallelizability. Parallelizability of long range interaction-based two-qubit gates is qualitatively compared to that of collisional gates. Different methods of performing single qubit gates are compared, and a lower bound of is determined on the error rate for the error mechanisms affecting Cs in a blue-detuned lattice with Raman transition-based single qubit gates, given reasonable limits on experimental parameters.

pacs:
03.67.Lx, 02.70.Hm, 37.10.Jk, 32.80.Qk

I Introduction

Neutral atoms trapped in optical lattices constitute a promising system for quantum information processing Deutsch and Jessen (1998); Deutsch et al. (2000). Single qubit operations and qubit readout have already been demonstrated Schrader et al. (2004), albeit in a non-scalable system, and a number of two-qubit gates have been proposed Jaksch et al. (2000, 1999). Addressable optical lattices—in which the lattice spacing is large enough that individual lattice sites can be targeted by a laser Scheunemann et al. (2000a)—offer an environment that can be scaled to thousands of qubits in a three-dimensional (3D) array Weiss et al. (2004). Preparation, loading, and imaging of an addressable optical lattice have recently been demonstrated Nelson et al. (2007).

Achieving large-scale fault-tolerant quantum computation requires single and two-qubit gates with extremely high fidelity, as well as the ability to perform many gates in parallel. Current estimates of the fault tolerance error threshold range from to for conventional quantum error correction, depending on the difficulty of communication between physically distant qubits and the ability to prepare certain states “offline” in a reliable manner Szkopek et al. (2006); Cheng and Silbey (2005); Aliferis et al. (2006); Svore et al. (2007); Steane (2003). More radical error correction schemes Raussendorf et al. (2006); Knill (2005a, b); Reichardt (2006) may offer better thresholds, but at the cost of high overhead. For the optical lattice system that is the focus of this work, the most relevant estimates Svore et al. (2007) suggest a fault tolerance threshold on the order of .

While much research has focused on schemes for realization of qubits and performing quantum operations in a wide variety of experimental systems, the detailed physics and scalability of specific architectures have to date received less attention. Some work has been done on system-level analysis of architectural issues Oskin et al. (2002); Copsey et al. (2003a, b); Isailovic et al. (2006). However, to provide real numbers to questions such as how large a system of qubits may be made and how many quantum operations can be made on this, it is necessary to undertake detailed analysis of the behavior of the proposed qubits in situ. The different physics involved in different implementations pose a variety of challenges, some of which may be system-specific while others, such as achieving gate fidelities with fault tolerant threshold values are quite generic. For example, the need to use thermal ensembles rather than pure states—as is usual in gas state and solid state proposals—has presented a major challenge for liquid state NMR because of limitations imposed by small thermal polarization and difficulties of initialization Warren et al. (1997); Braunstein et al. (1999); Jones (2001); Boykin et al. (2002).

For qubits defined in internal states of trapped neutral atoms or trapped ions, both the qubit interactions and their environmental decoherence mechanisms are well understood Deslauriers et al. (2006). This enables architectural issues to be examined with full microscopic analysis of all physical features of such qubits. For ion traps, analysis of the limiting features of trapped ion physics led to the proposal of a modular, multiplexed trap architecture to allow scale up from a few ( ) ions to many, possibly thousands of individually addressable ions Wineland et al. (1998); Kielpinski et al. (2002); Schaetz et al. (2004) and progress demonstrating components of such an architecture for small numbers of ions is now underway Rowe et al. (2002); Barrett et al. (2004); Hensinger et al. (2006); Stick et al. (2006); Seidelin et al. (2006). For neutral atoms trapped in optical lattices, up to 250 individual atomic qubits have been trapped and imaged in an optical lattice system that can be readily scaled up to include thousands of atoms and whose spacing is large enough to allow individual addressability Nelson et al. (2007). Large arrays of sublattice addressable trapped atoms have also been made Lee et al. (2007); Anderlini et al. (2007). Trapped neutral atoms and ions are somewhat complementary; individual addressing and quantum gates are more straightforward to implement with ions and have already been demonstrated experimentally for small numbers of ions (e.g., forming entangled states of ions Leibfried et al. (2005); Haffner et al. (2005) and implementing algorithms with two Brickman et al. (2005) and three qubits Chiaverini et al. (2004)) but achieving scaleup to even hundreds of ions still presents a serious technical challenge. In contrast, although scaleup to lattices containing hundreds of neutral atoms has been demonstrated in an addressable system Nelson et al. (2007) and gates have been demonstrated in non-addressable systems Anderlini et al. (2007) and in addressable dipole traps Yavuz et al. (2006); Jones et al. (2007), they have not yet been accomplished in lattices containing individually addressable atoms. Very recently, techniques for addressing individual sites in a lattice with small spacing have been proposed Gorshkov et al. (2008); Cho (2007).

In contrast to the situation for atomic and ionic qubits, progress in scaleup of solid state realizations of qubits—such as Josephson junctions or P in Si—from pairs to many qubits is in a far more rudimentary state. The kind of detailed analysis that is required to develop specific physical devices cannot be made yet, since the underlying microscopic physics of the qubits when in situ is not currently well enough understood, although significant progress is being made Clark et al. (2003). Furthermore, while solid state systems are often generically referred to as ‘scalable’ because of the ability to fabricate large scale solid state devices, the individual elements or qubits are not as reproducible as gas phase qubits due to the complexities and variability of their surroundings Keyes (2003). Nevertheless, architectural studies are beginning to be made for these systems Conrad et al. (2007); Hollenberg et al. (2006).

In this work, we go a step further in analysis of scaleup for trapped neutral atoms, undertaking a detailed physical investigation of the effects of single qubit errors and other imperfections that limit the scalability of neutral atom quantum computation in an individually addressable optical lattice. We consider different candidate single qubit gates and their sensitivity to various sources of experimental error. We then compare this to calculations of the threshold rate for fault tolerant computation under the appropriate conditions, in order to estimate how large a quantum computation may be made within current technological constraints and possible near-term improvements. To our knowledge, this is the first such detailed physical estimation of a practical limit on physical scaleup for any proposed experimental implementation of pure state quantum computation. We hope that this detailed analysis for trapped neutral atoms will spur similar analyses for other physical implementations once the relevant microscopic physics is better understood. Given that no experimental system will have unlimited scalability, such physical analysis of technical limits to scalable systems of functioning qubits within current technology is an essential complement to theoretical algorithmic scaling characteristics derived from complexity theory.

We restrict our analysis here to addressable optical lattices. While we do not explicitly consider the alternative lattices with global addressing that are also being studied experimentally Porto et al. (2003), we shall make some comments at the end of this paper on relative benefits that these other lattices might offer. The analysis in this paper employs a combination of perturbation theory and numerical techniques such as the pseudo-spectral method with a Chebychev decomposition of the Schrödinger propagator Kosloff (1994) to quantify the effects of both memory and gate errors deriving from all known sources for trapped atoms. In some respects our calculations complement and extend those of Saffman and Walker Saffman and Walker (2005) for Rb atoms in dipole traps. However, that work did not address the issue of scalability that is analyzed here after the various error rates have been quantified. When a specific choice of parameters is necessary, we consider here Cs atoms in an addressable optical lattice Nelson et al. (2007) of lattice constant m, with depth of . The lattice is ortho-rhombic in geometry, and is created by blue-detuned beams at 800 nm, with an intrapair angle of for each of the three pairs of beams. For the one-dimensional case, the lattice potential is given by:

(1)

Field-insensitive sub-levels of the hyperfine ground-state manifold are chosen as the qubit basis: and . The auxiliary levels and are also involved in the single qubit gate presented here (see Fig. 2). The procedure for loading and initializing the lattice is described in detail in Weiss et al Weiss et al. (2004), and Vala et al Vala et al. (2005a). We assume that atoms are cooled to the vibrational ground state, e.g., with 3D Raman sideband cooling Kerman et al. (2000); Han et al. (2000).

This rest of this paper is organized as follows. Section II.1 contains an analysis of the error mechanisms due to the lattice itself, such as scattering processes and loss of atoms from the lattice. Section II.2 analyzes the single qubit gate proposal based on a Raman two-photon process and Section II.3 analyzes microwave pulse-based single qubit gates. We consider the effects of off-resonant transitions of non-target atoms, scattering, heating of target atoms, and addressing beam targeting and intensity errors for both types of single qubit gate. Section II.4 contains a brief comparative discussion of the scalability of different classes of two-qubit gates. Section III provides an analysis of the results from the previous sections and their collective implications for the scalability of quantum computation in an addressable optical lattice. In section IV, we summarize our conclusions, discuss some possible ways to bypass the limitations identified here and identify some useful applications within the constraints established here.

Ii Error Mechanisms

Fault tolerance thresholds are sometimes expressed in terms of a “unified” error rate comprising both storage and gate errors, but are often also written in terms of separate gate and storage error rates (and occasionally also preparation and read-out error rates). In order for error correction to be able to keep up with storage errors, a practical quantum computer must be able to perform many gates in parallel. Consequently, estimates of error threshold values have typically assumed that gates can be performed on arbitrarily many qubits in parallel Steane (2003). Storage errors can then be considered on a similar footing to gate errors that occur with a frequency given by multiplying the storage error rate by the typical gate time, to obtain an effective ‘error per gate time’ that can be combined with the error per gate (EPG) in analysis of overall error rates.

Fault-tolerance threshold theorems assume maximal parallelizability Steane (2003), implying that all or nearly all qubits can be addressed simultaneously (, where is the total number of physical qubits and is the number that may be simultaneously addressed). In most proposed schemes for quantum computing this is extremely hard, except for the trivial case where one desires to perform the exact same gate on all atoms simultaneously. Parallelizability thus constitutes an important figure of merit, since if grows slower than , the effective storage error rate will eventually exceed the capacity of any error correction protocol. If only a fraction of qubits can be addressed, the effective EPG for a storage error will be approximately equal to the storage error rate multiplied by the ratio . As we show in Section III, is on the order of in 3D lattices, while in 2D lattices can be on the order of . is on the order of tens of microseconds for the microwave pulse-based single qubit gate, whereas with sufficient laser power, it can be nanoseconds or less for the Raman single qubit gate.

In the remainder of this section we derive expressions for the EPG for various decoherence mechanisms encountered by atoms trapped in an addressable optical lattice, using Cs as a specific example where necessary.

ii.1 Optical lattice-induced storage errors

We assume that the lattice has already been prepared, and that each lattice site is initially occupied by exactly one atom in the motional state. A detailed description of a procedure to achieve this perfectly loaded lattice is contained in Vala et al Vala et al. (2005a).

ii.1.1 Photon Scattering

Figure 1: (Color online) The solid blue line shows the Raman scattering cross-section for a Cs atom in a blue-detuned optical lattice, as a function of lattice light wavelength, . The dashed green line shows the frequency-dependent polarizability at the frequency . Both and are calculated for polarized light interacting with the initial state , with results for the state or opposite polarization light being similar.

Both Raman scattering, in which the initial and final states of the atom differ, and Rayleigh scattering, in which they do not, are sources of storage errors. Fortunately, the decohering effects of Rayleigh scattering can be partially suppressed with pulse sequences Andersen et al. (2004). For Raman scattering, no such method exists, and so we focus our analysis here on this form of scattering. The effective storage EPG due to Raman scattering is given by , where is the gate time and the scattering rate. Calculating the relative transition strengths (see Appendix the details), we find that roughly half of the errors induced by Raman scattering will be bit-flip errors, while the rest will be leakage to non-qubit states. The Raman cross section (where is the lattice light frequency) can be calculated as described in the Appendix, and is shown in Fig. 1 as a function of the lattice light wavelength. The scattering rate is related to the cross section by

(2)

where is average over an atomic spatial distribution of the peak electric field squared. We note that the optical lattice potential depth is given by 111See Eq. (16) in N. B. Delone and V. P. Krainov, Physics–Uspekhi 42, 669 (1999).. The polarizability is shown as a function of wavelength in Fig. 1. For an atom in the ground state in a red-detuned lattice , whereas in a blue-detuned lattice, , where is the characteristic trapping frequency. We can then calculate the Raman scattering rate for the blue-detuned and red-detuned cases:

(3a)
(3b)

Using Eq. (3a), we see that, for Cs in a blue-detuned optical lattice with the reference parameters given in Section I, we obtain a Raman scattering rate of , and thus an effective EPG value of .

ii.1.2 Qubit loss and leakage

Qubit loss errors are particularly serious, in that they can not be automatically corrected by error correcting codes. When an atom is lost from the optical lattice, or leaks into a non-qubit state, it is necessary to first detect the error before it can be corrected. The lost atom must be replaced before standard erasure error correcting codes Grassl et al. (1997) can be applied to correct the error. Detecting qubit loss requires that we have a method of detecting the presence of an atom at a given lattice site without disturbing its state.

Preskill Preskill (1998) identified a simple circuit for performing such loss detection measurements. The circuit requires an ancilla in a known state, two applications of a CNOT or CPHASE gate, a similar number of single-qubit gates, and a measurement of the ancilla. This measurement could fail by giving an incorrect result (false positives or false negatives), or by disturbing the state of the target atom. The latter type of error could be corrected by standard error correcting codes, while the former could be minimized by repeating the measurement as necessary. Another possibility for detecting qubit loss involves the use of a cavity QED system Vala et al. (2005b).

The need for having certain atoms in the lattice reserved for use as ancillas for this scheme could be avoided by transporting an extra-lattice ancilla atom where needed through the use of optical tweezers Kuhr et al. (2001). If an atom loss was detected, this ancilla would already be on hand to serve as a replacement. A drawback of this approach is that performing such operations in parallel would require many sets of optical tweezers. In the case of most leakage errors, parallelizable methods exist for detecting “leaked” atoms and returning them to a qubit state.

Fortunately, qubit loss rates are very low, with storage times as long as 25 s already reported Schrader et al. (2004). Collisions with background gas atoms are the primary cause of loss, and so it appears that storage times can be increased further through improved vacuum systems. It is also possible that a method may be found for performing loss detection measurements in parallel, which, when coupled with a means for replacing lost atoms, would allow qubit loss errors to be handled by standard error correction techniques. Consequently, qubit loss is not likely to be the dominant source of errors in the near future, and we will not consider it further in this paper.

ii.2 Raman-based single qubit gates

Two-photon Raman transitions present an attractive option for single qubit gates because of the associated speed of qubit manipulation. Raman-based single qubit rotations have recently been experimentally demonstrated on a time scale less than 100 ns for a single Rb atom trapped in an optical dipole trap  Jones et al. (2007). A theoretical analysis of factors contributing to gate imperfections for a single Rb atom concluded that gate fidelities of are possible Saffman and Walker (2005). We analyze here the error mechanisms arising during Raman gates implemented for Cs atomic qubits in a blue-detuned optical lattice.

We consider a Raman process in which the transition is driven with strength by -polarized light at a detuning of , and the transition is driven with strength by -polarized light at a detuning .

In the general case of a Raman-based single qubit gate with two-photon detuning , Rabi frequency , and pulse duration , we have an effective off-resonance Rabi frequency , and the rotation is approximately described by the following matrix Saffman and Walker (2005):

(4)

Unless otherwise noted, we assume zero two-photon detuning, i.e., . For the specific resonance case , the rotation matrix is as follows:

with and .

It is necessary to define a metric for fidelity of rotation operations. We consider a qubit in an arbitrary initial state undergoing a rotation , and compare it to a pulse, . The fidelity is then given by the following relation (5b):

(5a)
(5b)

where the overline represents an average over initial states and the corresponding error is given by . (Note that this definition differs from Saffman and Walker (2005) which considered the fidelity of a pulse on a specific initial state.)

ii.2.1 Neighbor atom errors

In the case of a single qubit gate performed with two orthogonal Raman lasers, an atom that is adjacent to the target atom and that is on the axis of one of the two lasers will experience a small undesired rotation. The effective Rabi frequency for this non-target atom is , where is given by Eq. (19). From Eq. (5a), we can determine the fidelity error in the desired identity operations for the four neighboring non-target atoms as:

where is the Raman laser wavelength

ii.2.2 Spontaneous emission

In the limit where , the probability of spontaneous emission during a pulse is , where is the natural lifetime of the state. For , the detuning is related to the Rabi frequency and Raman laser intensity by

(7)

therefore the probability of spontaneous emission is

(8)

ii.2.3 Raman beam AC Stark Shifts

The difference in AC Stark Shift between the logical and states gives rise to a phase shift . For a pulse, , so for polarized light we have . In the range , the ratio .

We now wish to calculate the variance due to atomic motion and spatial variation in the Raman beam intensity. Since, for typical parameters, the Rayleigh length of the Raman beam will be much larger than the beam waist (), we need consider only motion in the transverse direction. In the transverse direction at the beam waist, the intensity has the form , where is the intensity at the center, the beam waist, and the transverse distance from the center. The atomic motional states can be approximated by eigenstates of the two-dimensional harmonic trapping potential obtained by parabolic expansion of the transverse potential at the minima of the lattice potential.

For an atom in the resulting two-dimensional harmonic oscillator eigenstate , we calculate the variance of the phase shift using the fourth-order Taylor expansion of the Gaussian beam intensity 222Saffman and Walker in Ref. Saffman and Walker (2005) appear to consider an approximation of the form , which, while generally a reasonable approximation for a Gaussian beam, is less accurate in the center region () than the fourth-order Taylor expansion ., . This results in the variance

(In studying the temperature dependence of this effect, the reader may find it helpful to make the approximation .) From the expression for fidelity of a pulse, Eq. (5b), we see that the expected error probability will be .

ii.2.4 Atomic motion

In addition to the effects discussed above, atomic motion will introduce noise through variation in the effective pulse area, , and variation in the two-photon detuning, . The former effect is simply a result of atomic motion across the Gaussian profile of the Raman beams, and has a similar form to the result calculated in the previous section, Eq. (II.2.3). For a pulse, we obtain the following result for the variance:

This variation in pulse area will then result in an error

for a gate.

Doppler shifts of the Raman beams will cause variation in the two-photon detuning . Unlike the isolated two-site dipole trap situation considered by Saffman and Walker Saffman and Walker (2005), our system involves a 3D lattice and thus does not allow for a convenient first-order Doppler-free Raman laser configuration. We thus expect to see significant variation in the two-photon detuning due to atomic motion-induced Doppler shifts, as described by the following relation:

(11)
(12)

From the general expression for the rotation matrix, Eq. (4), we determine that this variation will result in a fidelity error

ii.2.5 Polarization effects

The Raman beams used to perform the single-qubit gate have a Gaussian profile. This means that, even at the beam waist, the beam will have a small component of polarization other than the desired , according to Saffman and Walker (2005):

(13)

where is the Rayleigh length. This extraneous polarization can result in leakage errors by causing transitions to states outside the computational basis. To estimate the probability of such errors, we determine the relative magnitude of the second term above, which corresponds to undesired polarization “seen” by a target atom in a vibrational state . Eq. (13) suggests that this can be estimated by the ratio of the spatial extent of the atom to the Rayleigh length, i.e., by

where is the characteristic trapping frequency in the harmonic approximation. There are four possible non-basis states into which the qubit could leak: . Since the matrix elements for the unwanted transitions are comparable to those for the desired transitions Steck (2003a), and the corresponding Rabi frequencies are smaller by a factor of , the probability of leakage into any particular state for a gate can be approximated by . The total leakage probability in the motional ground state, , is then four times that quantity, i.e.:

(15)

ii.2.6 Laser intensity noise and line-width

Noise in the Raman lasers affects the fidelity of the gate. If the relative intensity fluctuation is , then by Taylor-expanding eqn. (5b) with , we see this will result in an initial state-averaged fidelity error of for a gate and a gate.

Even if the Raman lasers are actively stabilized, shot noise provides a lower bound on relative intensity fluctuations. The fluctuation due to shot noise  Wineland et al. (1998) is given by

(16)

where is the quantum efficiency of the detector used in the stabilization circuit, is the frequency of the Raman laser, and is the laser power. If we assume and that our stabilization circuit reaches the lower bound, then the minimum fidelity error is

(17)

For detunings much smaller than the absolute optical frequency, we can use (i.e., the Raman transition frequency). For a Gaussian beam of power , the intensity at the waist is related to the power as , while , with given by (7). This results in the following estimate for laser intensity fluctuation-induced error:

(18)

ii.3 Microwave-based single qubit gates

Site-specific single-qubit gate operation in 3D lattices can also be achieved through the use of a far off-resonance addressing laser focused on a single lattice site, combined with pulsed global microwave fields Vala et al. (2005a); Scheunemann et al. (2000a, b); Zhang et al. (2006). In order to address a single atom, a Gaussian beam with waist substantially smaller than the lattice spacing is used, which results in the target atom seeing a much greater field than any neighboring atom. The intensity of a Gaussian beam is described by

(19)

where is the beam waist, the intensity at the center of the waist, is the Rayleigh length, and the beam width as a function of the axial coordinate .

The addressing beam causes an AC Stark Shift of the various levels of the target atom. Here we consider the scheme for Cs that is outlined in Figure 2. By choosing the “magic wavelength”, , for the addressing beam (for Cs, ), the and auxiliary levels receive AC Stark Shifts of equal magnitude but opposite sign, while the qubit levels and are unaffected. This allows the , , and transitions to be driven by global microwave pulses that are resonant only for the target atom. Alternatively, a collimated beam could be used to address an entire row of atoms (provided the row was not much longer than the Rayleigh length ), allowing identical operations to be performed in parallel. However, since the beam waist must be smaller than the lattice spacing , this implies that and consequently only a relatively small number of atoms can be addressed simultaneously using this method. We discuss this limitation further in the next subsection.

We have developed a software package, quantum simulation software (QSIMS) Beals (2004), for simulating the quantum dynamics of one and two-qubit gates in this and other systems. Using QSIMS, we discretize the spatial wavefunction of the atom on a grid, with a separate grid representing each possible internal state of the atom. Quantum dynamics are simulated by applying the Schrödinger propagator, expanded in Chebychev polynomials Kosloff (1994). The kinetic portion of the Hamiltonian is applied by means of a Fourier transform of the discretized wavefunction from the position basis to the momentum basis. Transitions between levels are treated with a dressed state approach Cohen-Tannoudji et al. (1992). Although QSIMS is capable of simulating three spatial dimensions, the symmetry of the system and the near-separability of the lattice potential make it reasonable in most cases to perform simulations in only one or two spatial dimensions. This results in a significant speed-up, since the run time of the simulations is for a grid of points.

We simulate the microwave pulse-based single qubit gate with QSIMS, using the parameters , , , , and (with microwave pulse intensity chosen appropriately to achieve this gate time). The atom is assumed to be initially in the qubit state and in the motional ground state, so the final state of the gate correspond to and the motional ground state. We investigated various different versions of the gate with QSIMS and found that the best performance is achieved by simultaneously driving all three transitions , , and , with pulse intensity chosen such that the strength of the first two transitions is , where is the strength of the transition.

Figure 2: (Color online) Schematics of a single qubit flip operation. A focused far off-resonant addressing laser produces an AC Stark Shift of the and levels in a single target atom. The qubit levels are the levels, labeled and . Levels and are auxiliary levels involved in the single qubit gate. (Most of the hyperfine sub-levels not involved in the gate are not shown here.) Three global microwave pulses tuned to the transitions , , and drive these transitions in the target atom with transition strengths , , and . Other atoms are not affected to first order (see below) since their transitions are not resonant. Our simulations indicate that best performance is achieved by simultaneously driving all transitions, with the relative transition strengths chosen such that .

ii.3.1 Off-resonant transitions

Since the microwave pulses proposed to perform these single qubit operations are applied globally, i.e., to the entire lattice, there is a small probability that such a pulse will cause a given non-target atom to undergo a non-resonant transition. We can minimize this probability by carefully tuning the gate parameters such that the pulse ends with non-target atoms in a local minima of their Rabi cycles, and by making the detuning, , large compared with the Rabi coupling . We are limited in our ability to do the former by our pulse timing resolution, , the uncertainty in pulse length .

Most qubits in the lattice will be far from the target qubit so that is small (Eq. (19)), and thus will not experience any Stark shift due to the addressing beam. The and transitions are detuned from these unshifted qubit transitions by , while the transition is detuned by . Since the probability of transition for any given atom is small, we can treat the transition amplitudes as independent, and calculate each transition probability using the Rabi formula, Eq. (20a), assuming that the coupling and pulse time are chosen such that and , where is the time required for one “leg” of the single qubit gate (see above and Figure 2). Since we wish to minimize for the purposes of reducing other types of errors discussed below, we note that the smallest value of for which the former condition is satisfied is . This results in the off-resonant transition probability estimates

(20a)
(20b)

With these estimates we can now ask, what is the corresponding EPG due to off-resonant transitions of all non-target atoms? If we can simultaneously address an entire row of atoms in a lattice of atoms, the EPG is . Unfortunately, this is challenging in even a modestly-sized lattice, as the intensity of the beam is inhomogeneous along the beam axis, on a length scale set by the Rayleigh length . By combining equations (19) and (20a), with , we estimate that the error for an atom at distance from the beam waist along the beam axis would be approximately . For typical parameter values, this limits us to addressing just a few lattice sites with a single beam before the error becomes large. In fact for our example parameters, the Rayleigh length is , and the error becomes of order unity at just one lattice site away from the beam waist.

If we cannot simultaneously address entire rows with a single addressing beam, the effective EPG will scale as . Although such scaling of an error mechanism would preclude scalable fault-tolerant quantum computation for an arbitrarily-large system, in practice it should not prove very restrictive for lattices with moderate numbers of qubits. For example, for microwave pulses of the appropriate frequency, can be on the order of one cycle, or s. For a single-qubit gate time of , this means a lattice of atoms could have an EPG of less than due to off-resonant transitions. Scaling implications are discussed further in Section III.

ii.3.2 Addressing beam-induced heating

Figure 3: A series of “snapshots” of the atomic spatial and internal state wavefunction for an atom in a 1D trapping potential undergoing a microwave pulse-based single qubit rotation gate, as simulated by QSIMS. The four relevant internal atomic states are labeled through , as in Fig. 2, and the curves represent the wavefunction for the atom’s center-of-mass coordinate. Simulation parameters have been chosen to exaggerate the gate errors so that they are visible in this figure. In (a), we see the initial state of the atom is the internal state (, ), with a Gaussian center-of-mass wavefunction. In (b) and (c), we see transitions to the auxiliary levels and . In (d) and (e), vibrationally-excited states become visible, particularly for the component of the total wavefunction in the state . In (f), we see the final state with a significant portion of the wavefunction not in the desired state, showing instead noticeable entanglement between spatial and internal degrees of freedom (e.g., the “bumpy” shape of the wavefunction component for the level).

The far off-resonant Gaussian addressing beam used to perform site-selective single qubit gates contributes harmonic trapping and anti-trapping potential terms for the and states, respectively, and also adds additional state-dependent anharmonic terms to the potential experienced by the atom. These anharmonic terms can generate entanglement between motional and internal degrees of freedom, as well as “heating” the atom to higher motional states. Perturbation theory shows that the most significant undesirable effect is due to the difference between the harmonic components of the trapping potentials experienced by atoms in the auxiliary states and , relative to those experienced in qubit states and .

The overlap between the vibrational ground state of the level and the first even vibrationally-excited state of the level is approximately , where is the characteristic trapping frequency defined above and is the addressing beam waist. From this, we can calculate the probability of transition to the first even vibrationally-excited state, using the fact that and :

Here we have simplified Eq. (II.3.2) by assuming that, since is of the same order as , the term will be of order unity.

We have tested these perturbative predictions by simulating the single-qubit gate with QSIMS as described above, with initial condition . The simulations show that the probability that the atom will not be in the motional ground state and desired qubit state after completion of the gate is . This is consistent with what is expected from application of Eq. (II.3.2) with our simulation parameters.

A small amount of heating does not in itself destroy the qubit state. However, as the vibrational energy of the atoms increases, the probability of other types of errors increases, and so the atoms will periodically need to be re-cooled to the ground state. Optical cooling can be done directly if the qubit state were first transferred to a different location, or could potentially be done with the qubit “in place” through a mechanism such as sympathetic cooling. Analysis of such re-cooling mechanisms is beyond the scope of this paper.

ii.3.3 Addressing beam-induced Raman Scattering

We now determine the Raman scattering rate for the target atom during a single-qubit microwave gate (Rayleigh scattering at this wavelength is negligible). Using Eq. (2), we can find the scattering rate in terms of . For polarized light at the magic wavelength , calculation using Eq. (27) yields a polarizability of for Cs in the or state (note that the polarizabilities for these states have opposite signs and that the polarizabilities for the states are essentially zero at ). Since the AC Stark shifts of the target atom and levels are given by , we can express the scattering rate in terms of as follows:

We then obtain the corresponding Raman scattering error per gate, , by multiplying the scattering rate of Eq. (II.3.3) by the single-qubit gate time .

ii.3.4 Addressing beam position error

If the addressing beam is off-target by an amount , Taylor expansion of (19) shows that the energy of the and levels will be shifted by an amount . This decreases the transition probability according to:

(23)

There is also an effect due to the perturbation of the eigenstates of the and states. The matrix element between the unperturbed harmonic oscillator ground state and the perturbed first excited state is . Assuming , the probability of exciting to a higher motional state during the or transitions is then:

For typical parameter values, this second effect is of greater significance and we will neglect the former in comparison with this. Note also that we have calculated the error only for one “leg” (i.e., transition) out of the three that compose the gate, and that the total error for the gate may be greater.

We have simulated the one-qubit microwave pulse-based gate between and using QSIMS with an addressing beam position error . We find that on completion of the gate, the probability that the atom will not be in the motional ground state and desired qubit state , is . Using Eq. (II.3.4) with the same parameters as this simulation yields a value . This is an estimate of the error in the and legs of the gate. For the leg, we replace (because the perturbation due to the addressing beam has opposite sign for the state versus the state), to obtain an error of . Summing these three errors, we obtain an overall error estimate of for the complete gate, which is within a factor of 3 of the value obtained from the simulation.

ii.4 Two-qubit gates

In this section, we make a qualitative comparison of the two-qubit gate techniques most commonly mentioned in the literature, with an emphasis on analysis of their different implications for scalability. Most schemes either involve stationary qubits and long-range interactions, or movable qubits and short-range collisional interactions between qubits.

ii.4.1 Long-range interaction-based gates

As with single qubit gates, our choice of a large lattice spacing allows for two-qubit operations to be performed in a site-specific manner. If the atoms are to remain stationary, a long-range interaction is required to perform a two-qubit gate. Furthermore, the interaction must somehow be controllable. Dipole-dipole interactions between atoms excited into Rydberg states are a promising candidate, and many variations on this idea have been proposed Jaksch et al. (2000); Saffman and Walker (2005); Lukin et al. (2001); Protsenko et al. (2002); Safronova et al. (2003); Ryabtsev et al. (2005); Cozzini et al. (2006); Brennen et al. (2000).

In one version of the Rydberg gate Jaksch et al. (2000), two nearby atoms are conditionally excited into Rydberg states via a coherent process Cubel et al. (2005). If both atoms were initially in the state, they are both excited into the Rydberg state where they interact via a dipole-dipole interaction, resulting in accumulation of a phase on the two-qubit state. They are then de-excited via another series of Raman pulses. Since the interactional phase accumulation occurs only in the case where both atoms are initially in the state, this is effectively a CPHASE gate. Estimates of both the speed and the maximum possible fidelity for Rydberg gates are reasonably promising Saffman and Walker (2005); Jaksch et al. (2000), with some putting the error rates achievable on the order of to  Cozzini et al. (2006). However, because of the inherent long-range nature of the interactions in this gate, when running these gates between multiple pairs of qubits in parallel, it is essential to take into account the effects of cross-talk between different qubit pairs, i.e., the dipole-dipole interactions between qubits from different pairs. To make a rough estimate of the degree of parallelization possible in the presence of such cross-talk, consider a three-dimensional lattice of atoms. Suppose an external static electric field is applied to induce a “permanent” dipole moment in the Rydberg-state atoms Jaksch et al. (2000): the level shift due to the resulting dipole-dipole interaction falls off as . The fidelity error due to cross-talk will therefore scale roughly as , where is the distance between different pairs of atoms that are simultaneously involved in Rydberg gate operations. We take the fidelity of a Rydberg gate performed between atoms one lattice spacing apart as unity, for reference. Then, taking a value of for the fault tolerance threshold for gate errors (a value intermediate between different estimates of the threshold for the case of local gates Szkopek et al. (2006); Svore et al. (2007)), this implies that the two-qubit gate cannot simultaneously be performed on multiple pairs of atoms within approximately ten lattice sites of each other. In a three-dimensional lattice, this geometric constraint limits us to simultaneously performing approximately one two-qubit gate per several hundred atoms. This in turn implies that the storage error rate during the two-qubit gate time would have to be about two orders of magnitude lower than this fault tolerance threshold (i.e., for the above example), to avoid accumulating additional idle time errors on the qubits not involved in the gates.

It is possible to mitigate this cross-talk limitation by using an interaction with more limited range, such as the van der Waals interaction present between Rydberg atoms when there is no hybridizing static electric field  Gallagher (2005). In general, interaction strengths for the zero external field case scale as (as is typical for van der Waals-type interactions), although there are exceptions Reinhard et al. (2007). This implies that cross talk errors would scale as , which allows for roughly one simultaneous two-qubit gate per three dozen atoms. This imposes more modest constraints on storage error rates—the storage error rate per two qubit gate time would then only need to be about one order of magnitude below the threshold value of .

ii.4.2 Collisional gates

Another method of avoiding the limitations of long-range interactions is to bring pairs of atoms close together and use short-range interactions or collisions to implement two-qubit gates. A variety of such gates have been proposed and analyzed Jaksch et al. (1999); Calarco et al. (2000); Stock et al. (2003); Mandel et al. (2003); Vager et al. (2005); Sebby-Strabley et al. (2006); Joo et al. (2006); Stock and Deutsch (2006); Idziaszek and Calarco (2006).

We note that, in the context of the system discussed in this paper, it would first be necessary to transfer atoms from the qubit states to states that experience a state-selective trapping potential (i.e., states with non-zero ), so as to allow pairs of atoms to be translated towards each other and brought together. For example, for lattice light with and a particular circular polarization, the states and have polarizabilities that differ by more than 8 %. Specific atoms can be transferred into these states by microwave or Raman transitions, analogous to the single qubit gates discussed earlier. Alternatively, an entire plane of atoms can be transferred simultaneously using the microwave pulse method if the addressing beam is replaced by an inhomogeneous magnetic field. Once the atoms are in the appropriate states, the atoms can be selectively moved by changing the relative polarization of one of the lattice beams, as described by Vala et al Vala et al. (2005a) and Weiss et al Weiss et al. (2004), allowing the atoms to be brought together to perform a gate Jaksch et al. (1999).

Due to the necessity of physically moving atoms around the lattice, collisional gates are likely to be much slower than long-range interaction-based gates. Even in the case of a “fast approach”, where the translation of the atoms is not adiabatic, gate times are still one to two orders of magnitude slower than the characteristic trap period of the lattice site’s potential well Vager et al. (2005). Estimates of the maximum fidelity possible with collisional gates also tend to be lower than maximum fidelity estimates for Rydberg gates Calarco et al. (2000).

Despite these drawbacks, collisional gates appear more suited to large-scale quantum computation than long-range interaction-based gates. Collisional gates can easily be performed on a massively parallel scale, particularly when used in the context of cluster-state quantum computing Raussendorf and Briegel (2001); Nielsen and Dawson (2004). In cluster-state (also known as “one-way”) quantum computing, two qubit gates are used only in the initial preparation of a large entangled “cluster” state. The computation itself is then effected via single-qubit measurements in a variety of bases, or equivalently, single qubit rotations followed by measurement in a particular basis. One recent scheme for cluster-state quantum computing offers fault tolerance thresholds as high as for local depolarizing errors and if there are also errors in preparation, gates, storage, and measurement Raussendorf et al. (2006). One can easily imagine building such a cluster state with atoms in a three-dimensional lattice by performing collisional gates in parallel on entire planes of atoms at a time.

Iii Analysis

Source Section EPG
Raman scattering (blue-detuned lattice) II.1.1
Raman scattering (red-detuned lattice) II.1.1
Neighbor atom errors (Raman gate): P II.2.1
Spontaneous emission (Raman gate): P II.2.2
AC Stark Shifts (Raman gate): P II.2.3
Atomic motion-reduced pulse area (Raman gate): P II.2.4
Detuning Doppler shift (Raman gate): P II.2.4
Polarization effects (Raman gate): P II.2.5
Laser intensity noise (Raman gate): P II.2.6
Table 1: Error Per Gate (EPG) due to various sources. It is assumed that atoms are in the vibrational ground state (i.e., ). The first two sources listed produce storage errors, while the rest cause gate errors.
Source Section EPG Numerical EPG
Raman scattering (blue-detuned lattice) II.1.1
Off-resonant transitions (microwave gate): P II.3.1
Addressing beam-induced heating (microwave gate): P II.3.2
Raman scattering (microwave gate): P II.3.3
Addressing beam position (microwave gate): P II.3.4
Table 2: Error Per Gate (EPG) due to various sources. In calculating a numerical value of the EPG, we used the parameters , with the other parameters as defined as in Sec. II.3 () Where indicated above by (