Circuit QED: single-step realization of a multiqubit controlled phase gate with one microwave photonic qubit simultaneously controlling n &#x02212; 1 microwave photonic qubits

Biaoliang Ye; Zhen-Fei Zheng; Yu Zhang; Chui-Ping Yang

doi:10.1364/OE.26.030689

1. Introduction

Multiple qubit gates play important roles and are a crucial element in quantum information processing (QIP). A multiqubit gate can in principle be decomposed into a series of two-qubit and single-qubit gates, and thus can be constructed by using these basic gates. However, it is commonly recognized that building a multiqubit gate is difficult via the conventional gate-decomposition protocol. This is because the number of basic gates, required for constructing a multiqubit gate, increases drastically as the number of qubits increases. As a result, the gate operation time would be quite long and thus the gate fidelity would be significantly decreased by decoherence. Hence, it is worthwhile to seek efficient approaches to realize multiqubit quantum gates. Many efficient schemes have been presented for the direct realization of a multiqubit controlled-phase or controlled-NOT gate, with multiple-control qubits acting on one target qubit [1–14]. This type of multiqubit gate is of significance in QIP, such as quantum algorithms and error corrections.

In this work, we focus on another type of multiqubit gate, i.e., a multi-target-qubit controlled phase gate with one qubit simultaneously controlling multiple target qubits. This multi-target-qubit controlled phase gate is described by

\begin{array}{l} | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 \to | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉, \\ | 1_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 \to | 1_{1} 〉 {(- 1)}^{i_{2}} {(- 1)}^{i_{3}} \dots {(- 1)}^{i_{n}} | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉, \end{array}

where i₂, i₃, ...i_n ∈ {0, 1} ; subscript 1 represents the control qubit while subscripts 2, 3, . . ., and n represent target qubits. From Eq. (1), it can be seen that when the control qubit 1 is in |1〉, a phase flip (from sign + to −) happens to the state |1〉 of each of target qubits 2, 3, . . ., and n ; however nothing happens to the states of each of target qubits 2, 3, . . ., and n when the control qubit 1 is in the state |0〉.

This multiqubit gate (1) is useful in QIP, such as entanglement preparation [15], error correction [16], quantum algorithms [17], and quantum cloning [18]. How to efficiently implement this multiqubit gate becomes necessary and important. Over the past years, based on cavity QED or circuit QED, many efficient methods have been proposed for the direct implementation of this multiqubit phase gate, by using natural atoms or artificial atoms (e.g., superconducting qubits, quantum dots, or nitrogen-vacancy center ensembles) [19–23].

Circuit QED is analogue of cavity QED, which consists of superconducting qubits and microwave resonators or cavities. It has developed fast recently and is considered as one of the most promising candidates for QIP [23–29]. Owing to the microfabrication technology scalability, individual qubit addressability, and ever-increasing qubit coherence time [30–38], superconducting qubits are of great importance in QIP. The strong and ultrastrong couplings between a superconducting qubit and a microwave cavity have been experimentally demonstrated [39,40]. For a review on the ultrastrong coupling, refer to [41]. On the other hand, a (loaded) quality factor Q ∼ 10⁶ has been experimentally reported for a one-dimensional coplanar waveguide microwave resonator [42,43], and a (loaded) quality factor Q ∼ 3.5× 10⁷ has also been experimentally reported for a three-dimensional microwave cavity [44]. A microwave resonator or cavity with the experimentally-reported high quality factor here can act as a good quantum data bus [45–47] and be used as a good quantum memory [48,49], because it contains microwave photons whose lifetimes are much longer than that of a superconducting qubit [50]. These good features make microwave resonators or cavities as a powerful platform for quantum computation and microwave photons as one of promising qubits for QIP. Recently, quantum state engineering and QIP with microwave fields or photons have become considerably interesting [51–72].

Motivated by the above, we will propose a method to realize the multi-target-qubit controlled phase gate (1) with microwave photonic qubits, by using n microwave cavities coupled to a superconducting flux qutrit (a Λ-type three-level artificial atom) (Fig. 1). Note that to simplify the presentation, we will use “MP qubit” to denote “microwave photonic qubit” and “MP qubits” to define “microwave photonic qubits”. This work is based on circuit QED. As shown below, this proposal has the following advantages: (i) During the gate operation, the qutrit stays in the ground state and thus decoherence from the qutrit is greatly suppressed; (ii) Because of only using one-step operation, the gate implementation is quite simple; (iii) Neither classical pulse nor measurement is required; (iv) The gate operation time is independent of the number of the qubits; and (v) This proposal is quite general and can be extended to a wide range of physical systems to realize the proposed gate, such as multiple microwave or optical cavities coupled to a natural or artificial Λ-type three-level atom. To the best of our knowledge, this work is the first to show the one-step implementation of a multi-target-qubit controlled phase gate with MP qubits, based on cavity- or circuit-QED and without any measurement.

Fig. 1 (a) Diagram of n cavities (1, 2, ..., n) coupled to a superconducting flux qutrit A. A square represents a cavity, which can be a one-dimensional or three-dimensional cavity. The qutrit is capacitively or inductively coupled to each cavity. (b) Level configuration of the flux qutrit, for which the transition between the two lowest levels can be made weak by increasing the barrier between two potential wells. (c) Diagram of a flux qutrit, which consists of three Josephson junctions and a superconducting loop.

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This paper is organized as follows. In Sec. 2, we explicitly show how to realize a controlledphase gate with one MP qubit simultaneously controlling n − 1 target MP qubits. In Sec. 3, we discuss the experimental feasibility for implementing a three-qubit controlled phase gate, by considering a setup of three one-dimensional transmission line resonators coupled to a superconducting flux qutrit. We end up with a conclusion in Sec. 4.

2. Multi-Target-Qubit controlled phase gate

Consider n microwave cavities (1, 2, ..., n) coupled to a superconducting flux qutrit [Fig. 1(a)]. The three levels of the qutrit are denoted as |g〉, |e〉 and |f〉, as shown in Fig. 1(b). In general, there exists the transition between the two lowest levels |g〉 and |e〉, which, however, can be made to be weak by increasing the barrier between the two potential wells. Suppose that cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling constant g₁ and detuning δ₁ but highly detuned (decoupled) from the |e〉 ↔ |f〉 transition of the qutrit. In addition, assume that cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling constant g_l and detuning δ_l but highly detuned (decoupled) from the |g〉 ↔ |f〉 transition of the qutrit (Fig. 2). Note that these conditions can be satisfied by prior adjustment of the qutrit’s level spacings or/and the cavity frequency. For a superconducting qutrit, the level spacings can be rapidly (within 1–3 ns) tuned by varying external control parameters (e.g., magnetic flux applied to the loop of a superconducting phase, transmon [73], Xmon [33], or flux qubit/qutrit [74]). In addition, the frequency of a microwave cavity or resonator can be rapidly adjusted with a few nanoseconds [75,76].

Fig. 2 Cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling strength g₁ and detuning δ₁, while cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling strength g_l and detuning δ_l. The purple vertical line represents the frequency ω_c₁ of cavity 1, while the blue, green, ..., and red vertical lines represent the frequency ω_c₂ of cavity 2, the frequency ω_c₃ of cavity 3,..., and the frequency ω_{c_n} of cavity n, respectively.

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Under the above assumptions and by considering the ideal cavities, the Hamiltonian of the whole system, in the interaction picture and after making the rotating-wave approximation (RWA), can be written as (in units of ħ = 1)

H_{I} = g_{1} (e^{- i δ_{1} t} {\hat{a}}_{1}^{+} σ_{f g}^{-} + h . c .) + \sum_{l = 2}^{n} g_{l} (e^{- i δ_{l} t} {\hat{a}}_{l}^{+} σ_{f e}^{-} + h . c .),

where

σ_{f g}^{-} = | g 〉 〈 f |

,

σ_{f e}^{-} = | e 〉 〈 f |

, δ₁ = ω_fg − ω_c₁ > 0, and δ_l = ω_fe − ω_{c_l} > 0. The detunings δ₁ and δ_l have a relationship δ_l = δ₁ + Δ_1l, with Δ_1l = ω_c₁ − ω_{c_l} − ω_eg > 0 (Fig. 2). Here, â₁ (â_l) is the photon annihilation operator of cavity 1 (l), ω_{c_l} is the frequency of cavity l (l = 2, 3, ..., n); while ω_fg, ω_fe, and ω_eg are the |f〉 ↔ |g〉, |f〉 ↔ |e〉, and |e〉 ↔ |g〉 transition frequencies of the qutrit, respectively.

Under the large-detuning conditions δ₁ ≫ g₁ and δ_l ≫ g_l, the Hamiltonian (2) becomes [77]

\begin{array}{l} H_{e} = & - λ_{1} ({\hat{a}}_{1}^{+} {\hat{a}}_{1} | g 〉 〈 g | - {\hat{a}}_{1} {\hat{a}}_{1}^{+} | f 〉 〈 f |) \\ - \sum_{l = 2}^{n} λ_{l} ({\hat{a}}_{l}^{+} {\hat{a}}_{l} | e 〉 〈 e | - {\hat{a}}_{l} {\hat{a}}_{l}^{+} | f 〉 〈 f |) \\ - \sum_{l = 2}^{n} λ_{1 l} (e^{i Δ_{1 l} t} {\hat{a}}_{1}^{+} {\hat{a}}_{l} σ_{e g}^{-} + h . c .) \\ + \sum_{k \neq l; k, l = 2}^{n} λ_{k l} (e^{i Δ_{k l} t} {\hat{a}}_{k}^{+} {\hat{a}}_{l} + h . c .) (| f 〉 〈 f | - | e 〉 〈 e |), \end{array}

where

λ_{1} = g_{1}^{2} / δ_{1}

,

λ_{l} = g_{l}^{2} / δ_{l}

, λ_1l = (g₁g_l/2) (1/δ₁ + 1/δ_l), λ_kl = (g_kg_l/2) (1/δ_k + 1/δ_l), Δ_1l = δ_l − δ₁ = ω_c₁ − ω_{c_l} − ω_eg, Δ_kl = δ_l − δ_k = ω_{c_k} − ω_{c_l}, and

σ_{e g}^{-} = | g 〉 〈 e |

. In Eq. (3), the terms in the first two lines describe the photon number dependent stark shifts of the energy levels |g〉, |e〉 and |f〉; the terms in the third line describe the |e〉 ↔ |g〉 coupling caused due to the cooperation of cavities 1 and l; while the terms in the last line describe the coupling between cavities k and l. For Δ_1l ≫ {λ₁, λ_l, λ_1l, λ_kl}, the effective Hamiltonian H_e changes to [77]

\begin{array}{l} H_{e} = & - λ_{1} ({\hat{a}}_{1}^{+} {\hat{a}}_{1} | g 〉 〈 g | - {\hat{a}}_{1} {\hat{a}}_{1}^{+} | f 〉 〈 f |) \\ - \sum_{l = 2}^{n} λ_{l} ({\hat{a}}_{l}^{+} {\hat{a}}_{l} | e 〉 〈 e | - {\hat{a}}_{l} {\hat{a}}_{l}^{+} | f 〉 〈 f |) \\ - \sum_{l = 2}^{n} χ_{1 l} ({\hat{a}}_{1}^{+} {\hat{a}}_{1} {\hat{a}}_{l} {\hat{a}}_{l}^{+} | g 〉 〈 g | - {\hat{a}}_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{l}^{+} {\hat{a}}_{l} | e 〉 〈 e |) \\ + \sum_{k \neq l; k, l = 2}^{n} λ_{k l} (e^{i Δ_{k l} t} {\hat{a}}_{k}^{+} {\hat{a}}_{l} + h . c .) (| f 〉 〈 f | - | e 〉 〈 e |), \end{array}

where

χ_{1 l} = λ_{1 l}^{2} / Δ_{1 l}

H_{e} = - λ_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} | g 〉 〈 g | - \sum_{l = 2}^{n} χ_{1 l} {\hat{a}}_{1}^{+} {\hat{a}}_{1} {\hat{a}}_{l} {\hat{a}}_{l}^{+} | g 〉 〈 g | .

Note that

[a_{l}, a_{l}^{+}] = 1

, i.e.,

{\hat{a}}_{l} {\hat{a}}_{l}^{+} = 1 + {\hat{a}}_{l}^{+} {\hat{a}}_{l}

. Thus, the Hamiltonian (5) can be rewritten as

H_{e} = - λ_{1} {\hat{n}}_{1} | g 〉 〈 g | - \sum_{l = 2}^{n} χ_{1 l} {\hat{n}}_{1} | g 〉 〈 g | - \sum_{l = 2}^{n} χ_{1 l} {\hat{n}}_{1} {\hat{n}}_{l} | g 〉 〈 g |,

where

{\hat{n}}_{1} = {\hat{a}}_{1}^{+} {\hat{a}}_{1}

and

{\hat{n}}_{l} = {\hat{a}}_{l}^{+} {\hat{a}}_{l}

are the photon number operators for cavities 1 and l, respectively.

Assume that the qutrit is initially in the ground state |g〉. It will remain in this state because the Hamiltonian (6) cannot induce any transition for the qutrit. Therefore, the Hamiltonian H_e reduces to

{\tilde{H}}_{e} = - η {\hat{n}}_{1} - χ \sum_{l = 2}^{n} {\hat{n}}_{1} {\hat{n}}_{l},

where η = λ₁ + (n − 1) χ. Here, we have set χ_1l = χ (l = 2, 3, ..., n). Note that the H̃_e is the effective Hamiltonian governing the dynamics of the n cavities (1, 2, ..., n).

The unitary operator U = e^−iH̃_et can be written as

U = U_{1} [\otimes_{l = 2}^{n} U_{1 l}],

with

U_{1} = exp (i η {\hat{n}}_{1} t),

U_{1 l} = exp (i χ {\hat{n}}_{1} {\hat{n}}_{l} t),

where

\otimes_{l = 2}^{n} U_{1 l} = U_{12} U_{13} \dots U_{1 n}

. Here, U₁ is a unitary operator on cavity 1, while U_1l is a unitary operator on cavities 1 and l.

Let us now consider n MP qubits 1, 2, ..., and n, which are hosted by cavities 1, 2, ..., and n, respectively. The two logical states of MP qubit l′ are represented by the vacuum state |0〉 and the single-photon state |1〉 of cavity l′ (l′ = 1, 2, ..., n). Based on Eq. (10), one can easily see that for χt = π, the unitary operation U_1l leads to the following state transformation

\begin{array}{l} U_{1 l} | 0_{1} 0_{l} 〉 = | 0_{1} 0_{l} 〉, \\ U_{1 l} | 0_{1} 1_{l} 〉 = | 0_{1} 1_{l} 〉, \\ U_{1 l} | 1_{1} 0_{l} 〉 = | 1_{1} 0_{l} 〉, \\ U_{1 l} | 1_{1} 1_{l} 〉 = - | 1_{1} 1_{l} 〉, \end{array}

which implies that the operator U_1l implements a universal controlled-phase gate on two qubits 1 and l. Eq. (11) can be expressed as

\begin{array}{l} U_{1 l} | 0_{1} i_{l} 〉 | g 〉 & = & | 0_{1} i_{l} 〉 | g 〉 \\ U_{1 l} | 1_{1} i_{l} 〉 | g 〉 & = & {(- 1)}^{i_{1}} | 1_{1} i_{l} 〉 | g 〉, \end{array}

where i_l ∈ {0, 1}.

Based on Eq. (12), one can easily obtain the following state transformation

\begin{array}{l} \otimes_{l = 2}^{n} U_{1 l} | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 = | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉, \\ \otimes_{l = 2}^{n} U_{1 l} | 1_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 = | 1_{1} 〉 {(- 1)}^{i_{2}} {(- 1)}^{i_{3}} \dots {(- 1)}^{i_{n}} | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 . \end{array}

According to (9), one can see that for ηt = 2mπ (m is a positive integer), the unitary operator U₁ leads to

U_{1} | 0_{1} 〉 = | 0_{1} 〉, U_{1} | 1_{1} 〉 = | 1_{1} 〉 .

Combining Eq. (13) and Eq. (14), we have

\begin{array}{l} U_{1} [\otimes_{l = 2}^{n} U_{1 l}] | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 = | 0_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉, \\ U_{1} [\otimes_{l = 2}^{n} U_{1 l}] | 1_{1} 〉 | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉 = | 1_{1} 〉 {(- 1)}^{i_{2}} {(- 1)}^{i_{3}} \dots {(- 1)}^{i_{n}} | i_{2} 〉 | i_{3} 〉 \dots | i_{n} 〉, \end{array}

which shows that when the control qubit 1 is in the state |1〉, a phase flip (from sign + to −) happens to the state |1〉 of each of target qubits (2, 3, ..., n), while nothing happens to the states of each of target qubit (2, 3, ..., n) when the control qubit 1 is in the state |0〉. From Eq. (8), it can be seen that the jointed unitary operators

U_{1} [\otimes_{l = 2}^{n} U_{1 l}]

involved in Eq. (15) is equivalent to the unitary operator U. By comparing Eq. (15) with Eq. (1), one can see that a multi-target-qubit controlled phase gate, described by Eq. (1), is realized with n MP qubits (1, 2, ..., n), after the above operation, described by the unitary operator U.

We stress that the gate is realized through a single unitary operator U, which was obtained by starting with the original Hamiltonian (2). In this sense, the gate is implemented with only a single operation. In addition, it is noted that the qutrit remains in the ground state |g〉 during the gate operation. Hence, decoherence from the qutrit is greatly suppressed.

In above, we have set χ_1l = χ, which turns out into

\frac{g_{1}^{2} g_{l}^{2}}{4 Δ_{1 l}} {(\frac{1}{δ_{1}} + \frac{1}{δ_{l}})}^{2} = χ .

In addition, we have set χt = π and ηt = 2mπ, from which we obtain

\frac{g_{1}^{2}}{δ_{1}} = (2 m - n + 1) χ .

Given g₁, δ₁, m, and n, the value of χ can be calculated based on Eq. (17). In addition, given g₁, δ₁, and χ, Eq. (16) can be satisfied by varying g_l or δ_l or both. Note that the detuning δ_l can be adjusted by varying the frequency of cavity l, and the coupling strength g_l can be adjusted by a prior design of the sample with appropriate capacitance or inductance between the qutrit and cavity l [13,78].

As shown above, the Hamiltonian (5) was obtained from the Hamiltonian (4) when the levels |e〉 and |f〉 are initially not occupied. This derivation has nothing to do with Δ_kl. In this sense, one can have Δ_kl ≠ 0 or Δ_kl = 0. Note that Δ_kl = δ_l − δ_k = ω_{c_k} − ω_{c_l}. Thus, the frequencies of cavities (2, 3, ..., n) can be chosen to be different or the same. However, it is suggested that for circuit QED, the frequencies of cavities should be different in order to suppress the unwanted inter-cavity crosstalk.

3. Possible experimental implementation

In this section, we briefly discuss the experimental feasibility of realizing a three-qubit controlled phase gate with one MP qubit simultaneously controlling two target MP qubits, by considering a setup of three microwave cavities (1, 2, 3) coupled to a superconducting flux qutrit (Fig. 3). Each cavity considered in Fig. 3 is a one-dimensional transmission line resonator (TLR).

Fig. 3 Setup for three one-dimensional transmission line resonators capacitively coupled to a superconducting flux qutrit.

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In reality, there exist the inter-cavity crosstalk between cavities [79], the unwanted coupling of cavity 1 with the |e〉 ↔ |f〉 transition, and the unwanted coupling of cavities 2 and 3 with the |g〉 ↔ |f〉 transition of the qutrit (Fig. 4). After taking these factors into account, the Hamiltonian (2) is modified as

{\tilde{H}}_{I} = H_{I} + δ H + ε,

with

\begin{array}{l} δ H & = & {\tilde{g}}_{1} (e^{- i {\tilde{δ}}_{1} t} {\hat{a}}_{1}^{+} σ_{f e}^{-} + h . c .) \\ + \sum_{l = 2}^{3} {\tilde{g}}_{l} (e^{- i {\tilde{δ}}_{l} t} {\hat{a}}_{l}^{+} σ_{f g}^{-} + h . c .), \end{array}

ε = \sum_{k \neq l; k, l = 1}^{3} g_{k l} (e^{i {\tilde{Δ}}_{k l} t} {\hat{a}}_{k}^{+} {\hat{a}}_{l} + h . c .) .

Here, H_I is the Hamiltonian (2) for n = 3. δH is the Hamiltonian, which describes the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition with coupling strength g̃₁ and detuning δ̃₁ = ω_fe − ω_c₁, as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition with coupling strength g̃_l and detuning δ̃_l = ω_fg − ω_{c_l} (l = 2, 3) (Fig. 4). In addition, ε represents the inter-cavity crosstalk, with the coupling strength g_kl between cavities k and l, as well as the frequency difference Δ̃_kl = ω_{c_k} − ω_{c_l} of cavities k and l (k ≠ l; k, l ∈ {1, 2, 3}).

Fig. 4 Illustration of the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition of the qutrit (with coupling strength g̃₁ and detuning δ̃₁) as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition of the qutrit (with coupling strength g̃_l and detuning δ̃_l) (l = 2, 3). Note that the coupling of each cavity with the |g〉 ↔ |e〉 transition of the qutrit is negligible because of the weak |g〉 ↔ |f〉 transition.

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When the dissipation and dephasing are included, the dynamics of the lossy system is determined by

\begin{array}{l} \frac{d ρ}{d t} & = & - i [{\tilde{H}}_{I}, ρ] + \sum_{l = 1}^{3} κ_{l} ℒ [a_{l}] \\ + γ_{e g} ℒ [σ_{e g}^{-}] + γ_{f e} ℒ [σ_{f e}^{-}] + γ_{f g} ℒ [σ_{f g}^{-}] \\ + \sum_{j = e, f} {γ_{φ j} (σ_{j j} ρ σ_{j j} - σ_{j j} ρ / 2 - ρ σ_{j j} / 2)}, \end{array}

where H̃_I is the above full Hamiltonian;

σ_{e g}^{-} = | g 〉 〈 e |

, σ_jj = |j〉〈j |(j = e, f); and

ℒ [ξ] = ξ ρ ξ^{†} - ξ^{†} ξ ρ / 2 - ρ ξ^{†} ξ / 2

, with ξ = a_l,

σ_{e g}^{-}

,

σ_{f e}^{-}

,

σ_{f g}^{-}

. In addition, κ_l is the photon decay rate of cavity l (l = 1, 2, 3), γ_eg is the energy relaxation rate for the level |e〉 of the qutrit, γ_fe(γ_fg) is the energy relaxation rate of the level |f〉 of the qutrit for the decay path |f〉 → |e〉(|g〉), and γ_φj is the dephasing rate of the level |j〉(j = e, f) of the qutrit.

The fidelity of the operation is given by

ℱ = \sqrt{〈 ψ_{id} | ρ | ψ_{id} 〉},

where |ψ_id〉 is the output state of an ideal system without dissipation, dephasing and crosstalk; while ρ is the final practical density operator of the system when the operation is performed in a realistic situation. For simplicity, we consider the three qubits are initially in the following state

\begin{array}{l} | ψ_{in} 〉 & = & \frac{1}{2 \sqrt{2}} (| 000 〉 + | 001 〉 + | 010 〉 + | 011 〉 \\ + | 100 〉 + | 101 〉 + | 110 〉 + | 111 〉) . \end{array}

Thus, the ideal output state of the whole system is

\begin{array}{l} | ψ_{id} 〉 & = & \frac{1}{2 \sqrt{2}} (| 000 〉 + | 001 〉 + | 010 〉 + | 011 〉 \\ + | 100 〉 - | 101 〉 - | 110 〉 + | 111 〉) \otimes | g 〉 . \end{array}

For a flux qutrit, the typical transition frequency between neighboring levels can be made as 1 to 20 GHz. As an example, we consider ω_eg/2π = 5.0 GHz, ω_fe/2π = 7.5 GHz, and ω_fg/2π = 12.5 GHz. By choosing δ₁/2π = 1.5 GHz, δ₂/2π = 1.51 GHz, and δ₃/2π = 1.53 GHz, we have Δ₁₂/2π = 10 MHz, Δ₁₃/2π = 30 MHz, ω_c₁/2π = 11 GHz, ω_c₂/2π = 5.99 GHz, and ω_c₃/2π = 5.97 GHz, for which we have Δ̃₁₂/2π = 5.01 GHz, Δ̃₂₃/2π = 0.02 GHz, and Δ̃₁₃/2π = 5.03 GHz. With the transition frequencies of the qutrit and the frequencies of the cavities given here, we have δ̃₁/2π = −3.5 GHz, δ̃₂/2π = 6.51 GHz, and δ̃₃/2π = 6.53 GHz. Other parameters used in the numerical simulation are: (i) $γ_{e g}^{- 1} = 5 T μ s$ , $γ_{f e}^{- 1} = 2 T μ s$ , $γ_{f g}^{- 1} = T μ s$ , (ii) $γ_{ϕ e}^{- 1} = γ_{ϕ f}^{- 1} = T μ s$ , and (iii) g₁/2π = 150 MHz. According to Eqs. (16) and (17), one can calculate the g₂ and g₃, which are g₂/2π ∼ 86.89 MHz and g₃/2π ∼ 151.49 MHz. For a flux qutrit, one has g̃₁ ∼ g₁, g₂ ∼ g₂, and g₃ ∼ g₃. Note that the coupling constants chosen here are readily available because a coupling constant ∼ 2π × 636 MHz has been reported for a flux device coupled to a one-dimensional transmission line resonator [40]. We set g_kl = 0.01g_max, where g_max = max{g₁, g₂, g₃} ∼ 2π × 151.49 MHz, which can be achieved in experiments [56,69]. In addition, assume κ₁ = κ₂ = κ₃ = κ for simplicity.

By solving the master equation (21), we numerically calculate the fidelity versus T and κ⁻¹, as depicted in Fig. 5. From Fig. 5, one can see that when T ⩾ 5 μs and κ⁻¹ ⩾ 10 μs, fidelity exceeds 0.9909, which implies that a high fidelity can be obtained for the gate being performed in a realistic situation.

Fig. 5 Fidelity versus T and κ⁻¹. The parameters used in the numerical simulation are referred to the text.

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To investigate the effect of the detuning errors on the fidelity, we consider a small deviation dδ for δ₁, δ₂, and δ₃. Thus, we modify δ₁, δ₂, and δ₃ as δ₁ + dδ, δ₂ + dδ, and δ₃ + dδ. With this modification, we numerically calculate the fidelity for T = 5 μs and κ⁻¹ = 10 μs and plot Fig. 6 showing the fidelity versus dδ. From Fig. 6, one can see that the fidelity can reach 0.98 or greater for −75 MHz ≤ dδ/2π ≤ 75 MHz.

Fig. 6 Fidelity versus dδ. Here, dδ is the detuning error, which applies to each of detunings δ₁, δ₂, and δ₃. The figure is plotted for T = 5 μs and κ⁻¹ = 10 μs. Other parameters used in the numerical simulation are the same as those used in Fig. 5.

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The gate operational time is estimated as ∼ 66.7 ns for the parameters chosen above, which is much shorter than the decoherence times of the qutrit (5 μs – 75 μs) and the cavity decay times (5 μs – 20 μs) considered in Fig. 5. Here, we consider a rather conservative case for decoherence time of the flux qutrit because decoherence time 70 μs to 1 ms has been experimentally reported for a superconducting flux device [32,36,38]. For the cavity frequencies given above and κ⁻¹ = 10 μs, one has Q₁ ∼ 6.9 × 10⁵ for cavity 1, Q₂ ∼ 3.76 × 10⁵ for cavity 2, and Q₃ ∼ 3.75 × 10⁵ for cavity 3, which are available because TLRs with a (loaded) quality factor Q ∼ 10⁶ have been experimentally demonstrated [42,43]. The analysis here implies that high-fidelity realization of a quantum controlled phase gate with one MP qubit simultaneously controlling two target MP qubits is feasible with the present circuit QED technology.

In above, we have provided the specific implementation of the three qubits case. For the gate with more than three qubits, the extension is straightforward. From Fig. 7, one can see that each of the multiple cavities can in principle be coupled to a single superconducting flux qutrit via a capacitor. However, it should be pointed out that in the solid-state setup scaling up to many cavities coupled to one qutrit will introduce new challenges. For instance, the cavity crosstalk may become worse as the number of cavities increases, which will decrease the operation fidelity.

Fig. 7 Schematic diagram for n cavities coupled by a superconducting flux qutrit. Each cavity here is a one-dimensional transmission line resonator, which is coupled to the qutrit via a capacitor.

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4. Conclusion

We have presented a one-step approach to realize an n-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling n − 1 target microwave photonic qubits, based on circuit QED. As shown above, this proposal has the following advantages: (i) During the gate operation, the qutrit remains in the ground state; thus decoherence from the qutrit is greatly suppressed; (ii) Because only one-step operation is needed and neither classical pulse nor measurement is required, the gate implementation is simple; (iii) The gate operation time is independent of the number of the qubits; and (iv) This proposal is quite general and can be applied to realize the proposed gate with a wide range of physical systems, such as multiple microwave or optical cavities coupled to a single Λ-type three-level natural or artificial atom. Furthermore, our numerical simulations demonstrate that high-fidelity implementation of a three-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling two target microwave photonic qubits is feasible with present circuit QED technology. We hope that this work will stimulate experimental activities in the near future.

Funding

NKRDP of China (2016YFA0301802); National Natural Science Foundation of China (11074062, 11374083,11774076).

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Circuit QED: single-step realization of a multiqubit controlled phase gate with one microwave photonic qubit simultaneously controlling n − 1 microwave photonic qubits

Abstract

1. Introduction

2. Multi-Target-Qubit controlled phase gate

3. Possible experimental implementation

4. Conclusion

Funding

References

Cited By

Figures (7)

Equations (24)

Optics Express