One-step implementation of Rydberg-antiblockade SWAP and controlled-SWAP gates with modified robustness

Jin-Lei Wu; Yan Wang; Jin-Xuan Han; Yu-Kun Feng; Shi-Lei Su; Yan Xia; Yongyuan Jiang; Jie Song

doi:10.1364/PRJ.415795

Abstract

The prevalent fashion of executing Rydberg-mediated two- and multi-qubit quantum gates in neutral atomic systems is to pump Rydberg excitations using multistep piecewise pulses in the Rydberg blockade regime. Here, we propose to synthesize a

Λ

-type Rydberg antiblockade (RAB) of two neutral atoms using periodic fields, which facilitates one-step implementations of SWAP and controlled-SWAP (CSWAP) gates with the same gate time. Besides, the RAB condition is modified so as to circumvent the sensitivity of RAB-based gates to infidelity factors, including atomic decay, motional dephasing, and interatomic distance deviation. Our work makes up the absence of one-step schemes of Rydberg-mediated SWAP and CSWAP gates and may pave a way to enhance the robustness of RAB-based gates.

The interaction between neutral atoms excited to Rydberg states is strong and long-range, making Rydberg atoms attractive in the context of quantum technologies [1 - 4]. Rydberg atoms have been considered as an attractive candidate platform for quantum computing [5 - 7] and quantum simulating [8 - 10] because of remarkable and continuous advances in cooling, trapping, and manipulating neutral atoms. Entangled states with scale up to 20 qubits have been generated in arrays of Rydberg atoms [11]. Furthermore, atomic species in experiments have been generalized from alkali metal atoms to alkaline earth atoms [12]. Although various schemes have been put forward to implement Rydberg-mediated quantum gates since the pioneering protocol was reported [13], enormous challenges remain in achieving experimentally high-fidelity Rydberg gates as well as in highly efficient Rydberg-atom-based quantum computing [1 - 4]. On the one hand, the gate fidelity is always limited due to intrinsic and technical errors. Intrinsic errors involve atomic decay and imperfect approximate conditions, including blockade errors in the Rydberg blockade [1, 14 - 16] gate schemes [13, 17 - 20], nonadiabatic errors [21, 22] in adiabatic gate schemes [13, 23 - 27], and higher-order perturbation errors in Rydberg antiblockade (RAB) [28 - 33] gate schemes [34 - 40]. Technical errors are caused by imperfections of techniques in, e.g., cooling, trapping, and manipulating atoms [1, 2, 41, 42]. On the other hand, existing schemes are not sufficient for one-step implementing certain two-qubit gates and many multi-qubit gates, especially for some frequently used gates, such as the SWAP gate, and the controlled-SWAP (CSWAP), that is, the Fredkin gate [43].

Despite a controlled-not (CNOT) gate combined with a small number of single-qubit gates constructing arbitrary gate operations (e.g., a SWAP gate formed with three CNOT gates [44]), direct executions of quantum gates can significantly improve the processing efficiency of lengthy quantum algorithms rather than decomposing them into a series of elementary gates [25, 45 - 48]. The SWAP gate is an important, nontrivial two-qubit gate with extensive applications in quantum computation [49], entanglement swapping [50], and quantum repeaters [51]. The CSWAP gate is one of the most representative multi-qubit gates, swapping the quantum states of two target qubits depending on the state of a control qubit, which holds important functions in quantum error correction [52], quantum fingerprinting [53], and quantum routers [54]. Among existing Rydberg-mediated gate schemes, SWAP gates are achieved in three or more steps, using multiple piecewise pulses and involving two or more Rydberg states in single atoms [55 - 57]. The scheme of implementing a CSWAP gate requires five-step operations with five piecewise pulses [58]. The multistep operations of implementing quantum gates not only make quantum algorithms rigmarole and unproductive but also accumulate more decoherence.

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Figure 1.(a) Schematic for implementing a SWAP gate. Two identical atoms are driven resonantly by two AM laser fields, excited from two ground (computational) states

| 0 ⟩

and

| 1 ⟩

to a Rydberg (mediated) state

| r ⟩

, respectively, with modulated Rabi frequencies

Ω_{0} (t)

and

Ω_{1} (t)

. Two atoms are coupled to each other by RRI with strength

V = C_{6} / d^{6}

C_{6}

being the van der Waals coefficient and

d

the interatomic distance. The effective

Λ

-type RAB dynamics is shown in the shadow of (b). (b) Schematic for implementing a CSWAP gate. Inset circle: the control atom

c

is coupled to target atoms 1 and 2 described in (a), with RRI strengths

V_{1 c}

and

V_{2 c}

corresponding to interatomic distances

d_{1 c}

and

d_{2 c}

, respectively. The effective

Λ

-type system of the target atoms is coupled to the control atom with RRI strength

(V_{1 c} + V_{2 c})

. In addition, the control atom is excited resonantly by another AM field from

{| 0 ⟩}_{c}

{| r ⟩}_{c}

with Rabi frequency

Ω_{c} (t)

Λ

\dot{ρ} = i [ρ, {\hat{H}}_{Full}] - \frac{1}{2} \sum_{j = 1}^{N} \sum_{k = 0}^{2} ({\hat{L}}_{k}^{j †} {\hat{L}}_{k}^{j} ρ - 2 {\hat{L}}_{k}^{j} ρ {\hat{L}}_{k}^{j †} + ρ {\hat{L}}_{k}^{j †} {\hat{L}}_{k}^{j}),

Figure 2.Time-dependent average fidelities of the SWAP gate with {

δ^{'} = 0

T = 3.87 μs

} and {

δ^{'} / 2 π = 1.11 MHz

T = 33.28 μs

}, respectively. Atomic decay is not considered.

Figure 3.Rydberg excitation probabilities during the SWAP gate procedure with different excitation numbers for (a) the resonant RAB with

δ^{'} = 0

and (b) the modified RAB with

δ^{'} / 2 π = 1.11 MHz

, respectively. Two-atom initial product state

| Ψ_{0} ⟩ = (| 0 ⟩_{1} + {| 1 ⟩}_{1}) / \sqrt{2} \otimes {| 1 ⟩}_{2}

is specified.

| r r ⟩

Figure 4.Infidelities of the SWAP gates caused by (a) atomic decay with different lifetimes of the Rydberg state, (b) motional dephasing with different atomic temperatures, (c) standard deviations of the interatomic distance, and (d) deviations in the RRI strength. Each point in (b), (c), and (d) denotes the average of 201 results.

Ω_{k} (t) \to Ω_{k} (t) e^{i Δ_{k} t}

d = \sqrt[6]{C_{6} / V}

Figure 5.Time-dependent average fidelities of the CSWAP gate with {

δ^{'} = 0

T = 3.87 μs

} and {

δ^{'} / 2 π = 1.11 MHz

T = 33.28 μs

}, respectively. Atomic decay is not considered.

Ω_{c m} / 2 π = 12 MHz

and

ω_{c} / 2 π = 142 MHz

, and

V_{1 c} / 2 π = V_{2 c} / 2 π = 70.98 MHz

Figure 6.Rydberg excitation probabilities during the CSWAP gate procedure with different excitation numbers for (a) the resonant RAB with

δ^{'} = 0

and (b) the modified RAB with

δ^{'} / 2 π = 1.11 MHz

, respectively. Three-atom initial product state

| Ψ_{0} ⟩ = (0 ⟩_{1} - {| 1 ⟩}_{1}) / \sqrt{2} \otimes {(| 0 ⟩}_{2} - {| 1 ⟩}_{2}) / \sqrt{2} \otimes {(| 0 ⟩}_{c} - {| 1 ⟩}_{c}) / \sqrt{2}

is specified.

Figure 7.Infidelities of the CSWAP gates caused by (a) atomic decay with different lifetimes of the Rydberg state, (b) motional dephasing with different atomic temperatures, (c) standard deviations of the distance between the two target atoms, and (d) deviations in the RRI strength between the two target atoms. Each point in (b), (c), and (d) denotes the average of 201 results.

Λ

With the two-atom basis {

| j k ?

} (

j, k = 0, ?? 1, ? r

), the full Hamiltonian of two atoms is

{\hat{H}}_{0} = \frac{1}{2} [Ω_{0 m} ? \cos (ω_{0} t) (| 00 ? ? r 0 | + | 01 ? ? r 1 | + | 0 r ? ? r r | + | 00 ? ? 0 r | + | 10 ? ? 1 r | + | r 0 ? ? r r |) + Ω_{1 m} ? \cos (ω_{1} t) (| 10 ? ? r 0 | + | 11 ? ? r 1 | + | 1 r ? ? r r | + | 01 ? ? 0 r | + | 11 ? ? 1 r | + | r 1 ? ? r r |) + H .c .] + V | r r ? ? r r | .

(A1)

We transform Eq.?(A1) into the frame defined by

{\hat{U}}_{0} = \exp (i t δ | r r ? ? r r |)

with

δ = V ? δ_{0} = ω_{1} ? ω_{0} ? δ_{0}

and obtain

{\hat{H}}_{1} = {\hat{U}}_{0} ({\hat{H}}_{0} ? i \frac{?}{? t}) {\hat{U}}_{0}^{?} = \hat{H}'_{1} + δ_{0} | r r ? ? r r |

with

{\hat{H}}_{1}^{'} = \frac{Ω_{0 m}}{4} {(e^{i ω_{0} t} + e^{? i ω_{0} t}) (| 00 ? ? r 0 | + | 01 ? ? r 1 | + | 00 ? ? 0 r | + | 10 ? ? 1 r |) + [e^{i (ω_{0} ? δ) t} + e^{? i ω_{1} t}] (| 0 r ? ? r r | + | r 0 ? ? r r |)} + \frac{Ω_{1 m}}{4} {(e^{i ω_{1} t} + e^{? i ω_{1} t}) (| 10 ? ? r 0 | + | 11 ? ? r 1 | + | 01 ? ? 0 r | + | 11 ? ? 1 r |) + [e^{i ω_{0} t} + e^{? i (ω_{1} + δ) t}] (| 1 r ? ? r r | + | r 1 ? ? r r |)} + H .c .

(A2)

When considering

| ω_{0} |, | ω_{1} |, | ω_{0} ? δ |, | ω_{1} + δ | ? | Ω_{0 m} | / 4

| Ω_{1 m} | / 4

, the terms

\frac{Ω_{0 m}}{4} (e^{i ω_{0} t} + e^{? i ω_{0} t}) (| 00 ? ? r 0 | + | 00 ? ? 0 r |) + \frac{Ω_{1 m}}{4} (e^{i ω_{1} t} + e^{? i ω_{1} t}) (| 11 ? ? r 1 | + | 11 ? ? 1 r |) + H .c.

can be neglected under the rotating-wave approximation because the transitions are of large detunings; besides, the involved even-parity states,

| 00 ?

and

| 11 ?

, cannot be effectively coupled resonantly to other states yet through two-photon processes. In addition, under the second-order perturbation theory, the even-parity computational states

| 00 ?

and

| 11 ?

have a zero-value sum of Stark shifts, and all single-excitation states are uncoupled to the four computational states. Then, the remaining part of Eq.?(A2) can be sorted as

{\hat{H}}_{1}^{'} ? [\frac{Ω_{0 m}}{4} (e^{i ω_{0} t} + e^{? i ω_{0} t}) | 01 ? ? r 1 | + \frac{Ω_{1 m}}{4} e^{i ω_{0} t} | r 1 ? ? r r |] + [\frac{Ω_{1 m}}{4} (e^{i ω_{1} t} + e^{? i ω_{1} t}) | 01 ? ? 0 r | + \frac{Ω_{0 m}}{4} e^{? i ω_{1} t} | 0 r ? ? r r |] + [\frac{Ω_{0 m}}{4} (e^{i ω_{0} t} + e^{? i ω_{0} t}) | 10 ? ? 1 r | + \frac{Ω_{1 m}}{4} e^{i ω_{0} t} | 1 r ? ? r r |] + [\frac{Ω_{1 m}}{4} (e^{i ω_{1} t} + e^{? i ω_{1} t}) | 10 ? ? r 0 | + \frac{Ω_{0 m}}{4} e^{? i ω_{0} t} | r 0 ? ? r r |] + \frac{Ω_{0 m}}{4} e^{i (ω_{0} ? δ) t} (| 0 r ? ? r r | + | r 0 ? ? r r |) + \frac{Ω_{1 m}}{4} e^{? i (ω_{1} + δ) t} (| 1 r ? ? r r | + | r 1 ? ? r r |) + H .c .

(A3)

With the second-order perturbation theory, the first four terms in Eq.?(A3) induce the effective coupling of

| 01 ? ? | r r ? ? | 10 ?

and Stark shifts of

| r r ?

, while the last two terms induce solely Stark shifts of

| r r ?

. Therefore, a final effective Hamiltonian of the two atoms can be obtained as

{\hat{H}}_{e} = [\frac{Ω_{e}}{2} (| 01 ? ? r r | + | 10 ? ? r r |) + H .c .] + δ^{'} | r r ? ? r r |,

(A4)

in which

Ω_{e} = Ω_{0 m} Ω_{1 m} / 8 ω_{1} ? Ω_{0 m} Ω_{1 m} / 8 ω_{0}

and

δ^{'} = Δ_{rr} + δ_{0}

with

Δ_{rr} = Ω_{0 m}^{2} / 8 ω_{1} ? Ω_{1 m}^{2} / 8 ω_{0} + Ω_{1 m}^{2} / 8 (ω_{1} + δ) ? Ω_{0 m}^{2} / 8 (ω_{0} ? δ)

being the sum Stark shift of

| r r ?

APPENDIX B: DEFINITION OF THE TRACE-PRESERVING-QUANTUM-OPERATOR-BASED AVERAGE FIDELITY

According to Nielsen's work [63], the trace-preserving-quantum-operator-based average fidelity of a quantum gate is defined as

\overset{ˉ}{F} (ε, \hat{U}) = {\sum_{j = 1}^{4^{N}} tr [\hat{U} {\hat{u}}_{j}^{?} {\hat{U}}^{?} ε ({\hat{u}}_{j})] + l^{2}} / l^{2} (l + 1),

(B1)

where

\hat{U}

is the ideal gate,

{\hat{u}}_{j} = ?_{k}^{N} {\hat{σ}}_{k}

is the tensor of Pauli matrices

{\hat{σ}}_{k} \in {\hat{I}, {\hat{σ}}^{x}, {\hat{σ}}^{y}, {\hat{σ}}^{z}}

on computational states {

| 0 ?

| 1 ?

}, and

l ?? {= 2}^{N}

for an

N

-qubit gate.

ε ({\hat{u}}_{j})

is a trace-preserving quantum operation obtained through our logic gates that can be solved by the master equation.

First, it is clear that the evolution from three-atom computational states

{| 001 ?}_{12 c}

and

{| 111 ?}_{12 c}

is prohibited. For

{| 000 ?}_{12 c}

{| 110 ?}_{12 c}

, the governing Hamiltonian is

{\hat{H}}_{β 0} = Ω_{c m} (e^{i ω_{c} t} + e^{? i ω_{c} t}) {| β 0 ?}_{12 c} ? β r | / 4 + H .c .

(

β = 00, ?? 11

). No evolution will occur when

| ω_{c} | ? | Ω_{c m} | / 4

is considered, because the laser-induced transitions are largely detuned, and the sum Stark shift of

{| β 0 ?}_{12 c}

is zero.

When the state of three atoms is

{| 010 ?}_{12 c}

{| 100 ?}_{12 c}

, the governing Hamiltonian of the three atoms is

{\hat{H}}_{2} = {\hat{H}}^{'}_{2} + δ^{'} {| r r ?}_{12} {? r r | + (V_{1 c} + V_{2 c}) | r r r ?}_{12 c} ? r r r |

with

{\hat{H}}_{2}^{'} = \frac{Ω_{c m}}{4} (e^{i ω_{c} t} + e^{? i ω_{c} t}) (| 010 ?_{12 c} {? 01 r | + | 100 ?}_{12 c} ? 10 r | + {| r r 0 ?}_{12 c} ? r r r |) + \frac{Ω_{e}}{2} {(| 010 ?}_{12 c} {? r r 0 | + | 100 ?}_{12 c} ? r r 0 | + {| 01 r ?}_{12 c} {? r r r | + | 10 r ?}_{12 c} ? r r r |) + H .c .

(C1)

Transforming

{\hat{H}}_{2}

to the frame defined by

{\hat{U}}_{1} = \exp ? {(i t ω_{c} | r r r ?}_{12 c} ? r r r |)

, one can obtain

{\hat{H}}_{3} = {\hat{H}}_{3}^{'} + δ^{'} {| r r ?}_{12} {? r r | + (V_{1 c} + V_{2 c} ? ω_{c}) | r r r ?}_{12 c} ? r r r |

with

{\hat{H}}_{3}^{'} = \frac{Ω_{c m}}{4} [(e^{i ω_{c} t} + e^{? i ω_{c} t}) (| 010 ?_{12 c} {? 01 r | + | 100 ?}_{12 c} ? 10 r |) + (1 + e^{? 2 i ω_{c} t}) {| r r 0 ?}_{12 c} ? r r r |] + \frac{Ω_{e}}{2} {(| 010 ?}_{12 c} ? r r 0 | + {| 100 ?}_{12 c} {? r r 0 | + | 01 r ?}_{12 c} {? r r r | e^{? i ω_{c} t} + | 10 r ?}_{12 c} ? r r r | e^{? i ω_{c} t}) + H .c .

(C2)

Through neglecting frequent oscillations under rotating-wave approximation with the condition

| ω_{c} | ? | Ω_{c m} | / 4, | Ω_{e} | / 2

{\hat{H}}_{3}^{'}

becomes

{\hat{H}}_{3}^{''} = [\frac{Ω_{c m}}{4} {| r r 0 ?}_{12 c} ? r r r | + \frac{Ω_{e}}{2} {(| 010 ?}_{12 c} {? r r 0 | + | 100 ?}_{12 c} ? r r 0 |) + H .c .] + Δ_{r r r} {| r r r ?}_{12 c} ? r r r |,

(C3)

with

Δ_{r r r} = Ω_{e}^{2} / 2 ω_{c} + Ω_{c m}^{2} / 32 ω_{c}

being the Stark shift of

{| r r r ?}_{12 c}

. In this case, an effective three-atom Hamiltonian of

{\hat{H}}_{3}

can be obtained as

{\hat{H}}_{4} = {\hat{H}}_{3}^{''} + δ^{'} {| r r ?}_{12} ? r r |

with the condition

V_{1 c} + V_{2 c} = ω_{c} ? Δ_{r r r}

{\hat{H}}_{4}

can be rewritten as

{\hat{H}}_{4} = [\frac{Ω_{e}}{\sqrt{2}} {(| 010 ?}_{12 c} + {| 100 ?}_{12 c}) (? ?_{0} | + ? ?_{1} |) + H .c .] + \frac{Ω_{c m}}{4} \sum_{n = 0}^{1} {(? 1)}^{n} | ?_{n} ? {? ?_{n} | + δ^{'} | r r ?}_{12} ? r r |,

(C4)

in which

| ?_{n} ? = ({| r r 0 ?}_{12 c} + {(? 1)}^{n} {| r r r ?}_{12 c}) / \sqrt{2}

. Transforming

{\hat{H}}_{4}

to the frame defined by

{\hat{U}}_{2} = \exp (i t \hat{A})

with

\hat{A} = Ω_{c m} \sum_{n = 0}^{1} {(? 1)}^{n} | ?_{n} ? {? ?_{n} | / 4 + δ^{'} | r r ?}_{12} ? r r |

, one can obtain a Hamiltonian with entirely off-resonant interactions

{\hat{H}}_{5} = \frac{Ω_{e}}{\sqrt{2}} {(| 010 ?}_{12 c} + {| 100 ?}_{12 c}) [? ?_{0} | e^{? i t (Ω_{c m} / 4 + δ^{'})} + ? ?_{1} | e^{i t (Ω_{c m} / 4 ? δ^{'})}] + H .c .

(C5)

When the condition

| Ω_{c m} / 4 \pm δ^{'} | ? | Ω_{e} | / \sqrt{2}

is satisfied, transitions from

{| 010 ?}_{12 c}

{| 100 ?}_{12 c}

can be banned.

Now that the evolution from six three-atom computational states, including

{| 000 ?}_{12 c}

{| 010 ?}_{12 c}

{| 100 ?}_{12 c}

{| 110 ?}_{12 c}

{| 001 ?}_{12 c}

, and

{| 111 ?}_{12 c}

, is banned, the dynamics of the three atoms can be governed by an effective Hamiltonian

{\hat{H}}_{eff} = [\frac{Ω_{e}}{2} {(| 011 ?}_{12 c} + {| 101 ?}_{12 c}) ? r r 1 | + H .c .] + δ^{'} {| r r 1 ?}_{12 c} ? r r 1 |,

(C6)

which is exactly Eq.?(8) in the main text.