Fast binomial-code holonomic quantum computation with ultrastrong light-matter coupling

We propose a protocol for bosonic binomial-code nonadiabatic holonomic quantum computation in a system composed of an artificial atom ultrastrongly coupled to a cavity resonator. In our protocol, the binomial codes, formed by superpositions of Fock states, can greatly save physical resources to correct errors in quantum computation. We apply to the system strong driving fields designed by shortcuts-to-adiabatic methods. This reduces the gate time to tens of nanoseconds. Decoherence of the system accumulated over time can be effectively reduced by such a fast evolution. Noise induced by control imperfections can be suppressed by a systematic-error-sensitivity nullification method, thus the protocol is mostly insensitive to such noises. As a result, this protocol can rapidly ($\sim 35$ ns) generate fault-tolerant and high-fidelity ($\gtrsim 98\%$ with experimentally realistic parameters) quantum gates.

Unfortunately, universal control of a single bosonic mode is difficult due to its harmonicity. Although adding direct and indirect nonlinear interactions can induce weak anharmonicity [53], it is still difficult to manipulate independently and simultaneously every needed Fock state. Moreover, weak nonlinear interactions may induce additional noises into the system and limit the gate fidelities [53]. This, with additional operations (e.g., feedback [60,65,66] and driven-dissipative controls [48,51]) and conditions (e.g., oscillators and qubits are never driven simultaneously [55,67,68]), makes it difficult to implement NHQC [14,17,18] with bosonic error-correction codes. Note that the first experiment for binomial-code conditional geometric gates was recently realized [55] using 3D superconducting cavities, but it is not a holonomic computation.

II. MODEL AND EFFECTIVE HAMILTONIAN
Our system consists of a three-level (|e , |g , |µ ) artificial atom and a cavity resonator [103]. The states |e and |g are ultrastrongly coupled to a cavity mode [104], with coupling strength g (see Fig. 1). The atom-cavity interaction is described by is the Rabi Hamiltonian. Here, σ x g = |e g| + |g e| and σ z g = |e e| − |g g| are Pauli matrices, a (a † ) is the annihilation (creation) operator of the cavity field, ω µ is the frequency of the level |µ , ω c,(q) is the cavity (qubit) frequency. In the USC regime (g/ω c 0.1), the eigenstates |E j with eigenvalues ξ j of H 0 can be separated into (i) noninteracting sectors |µ |n with eigenvalues ω µ + nω c ; and (ii) dressed atom-cavity states |ζ m with eigenvalues E m (j, n, m = 0, 1, 2, . . .). Here, |n denote the Fock states of the cavity mode, and denote the dressed states of H R . The coefficients c m n = ζ m |g |n and d m n±1 = ζ m |e |n ± 1 can be obtained numerically. Note that we impose d m −1 = 0 for Eq. (2). Oscillations |µ |n ↔ |ζ m can be induced by driving the atomic transition |µ ↔ |g [see Fig. 1

(b)] with an additional control Hamiltonian
Here, is a composite pulse [65,91,105,106] with amplitudes Ω k , frequencies ω k , and phases φ k . We omit the explicit time dependence of all the parameters (e.g., Ω k and φ k ) regarding the drivings. The total Hamiltonian is Choosing and performing a unitary transformation exp (−iH 0 t), we derive an effective Hamiltonian that, under the rotating wave approximation, is (see details in Appendix A) This effective Hamiltonian describes transitions between the Fock states |k through the dressed intermediate state |ζ m . We assume so that the dressed state |ζ m is the highest level in the evolution subspace. In Fig. 2(a), we illustrate the effective transitions for m = 0. Note that each Fock state can be freely populated by the drivings Ω k when the system is in the USC regime. Instead, in the weak-coupling regime, the qubit driving H D (t) only induces oscillations |g |0 ↔ |µ |0 because c m=0 n =0 0.

III. NONADIABATIC HOLONOMIC QUANTUM COMPUTATION VIA BINOMIAL CODES
An example of the binomial codes [54] for single-qubit gates protecting against the single-photon loss error is which form a computational subspace S c = {|0 , |1 }.
With this definition, a photon loss error brings the logical code words to a subspace with odd photon numbers that is clearly disjoint from the even-parity subspace of the logical code words [54]. The Knill-Laflamme condition [107,108] for this kind of codes reads ˜ |a † a|˜ = 2 ( , = 0, 1). This means that the probability of a photon jump to occur is the same for |0 and |1 , implying that the quantum state is not deformed under the error of a photon loss. For instance, when encoding quantum information as a photon jump leads to which means that the information (cos χ and sin χ) is not deformed [54].
To manipulate the codes in Eq. (8), we need a three-frequency composite pulse, i.e., k = (0, 2, 4) in Eq. (6). When g/ω c 0.5, the probability amplitudes c 2 0 , c 2 2 , c 2 4 of the Fock states (|0 , |2 , |4 ) in the third dressed state |ζ 2 are greater than in the other dressed states (see more details in Appendix A). For this reason, we choose m = 2 in Eq. (6). Assuming c 2 0 Ω 0 = c 2 4 Ω 4 and φ 0 = φ 4 , H eff (t) becomes an effective Λ-type system with two ground states {|0 , |1 } and an excited state |ζ 2 ≡ |ζ m=2 . The NHQC in a Λ-type system has been well studied [3,4,11]. For clarity, we define an effective driving amplitude and a time-independent parameter to rewrite H eff (t) to be Initially, quantum information is stored in the logical qubit states of the subspace S c (the atom is in |µ ). According to the invariant-based approaches for Λ-type transitions [100,101,109], when Ξ sin φ 2 = Ω p (β, ϕ) ≡ (β cot ϕ sin β +φ cos β), Ξ cos φ 2 = Ω s (β, ϕ) ≡ (β cot ϕ cos β −φ sin β), (15) the Hamiltonian in Eq. (13) can drive the system to evolve exactly along one of the two user-defined path (see details in Appendix B) which are two eigenstates of a dynamical invariant I(t) . For instance, when ϕ(0) = 0, the evolution is along |ψ − (t) , which acquires a dynamical phase and a geometric phase For a cyclic evolution, the time-dependent auxiliary parameters β and ϕ need to satisfy β(0) = β(t f ) and Thus, the final phases are ϑ − (t f ) = 0 and Θ − (t f ) = 2Θ s [see Fig. 2(b) and Appendix B], resulting in a geometric evolution. In the computational subspace S c , the evolution operator is (omitting a global phase Θ s ) This is a universal single-qubit gate. In this case, the gate time is T ∼ 18/c k 2 Ω peak k . In the USC regime we can assume c m k 0.1 and ω c /2π ∼ 5 GHz [72,73], resulting in T 5 ns, i.e., the gate time can be tens of nanoseconds. Choosing T = 35 ns, g 0.8ω c and ω c /2π = 6.25 GHz [104], the pulses Ω 0, (2,4) are shown in Fig. 3(a). Note there that the peak values of the pulses are Ω peak k /2π ∼ 200 MHz. These satisfy the condition ω c , g Ω k .

IV. ROBUSTNESS AGAINST CONTROL IMPERFECTIONS AND DECOHERENCE
It has been experimentally verified [110] that the pulses chosen based on β and ϕ in Eq. (19) can counteract the systematic errors induced by imperfections of the control fields Ω k , making the computation insensitive to such errors [18,19,102].
In the presence of such imperfections with error parameter δ i , the driving amplitudes become Ω i k = (1 + δ i )Ω k . Accordingly, the effective Hamiltonian H eff (t) should be corrected as H i eff (t) = (1 + δ i )H eff (t). By using time-dependent perturbation theory up to O(δ i ), the evolution state of the system is approximatively where U (t f , t) is the unperturbed time evolution operator.
We assume that the protocol works perfectly when δ i = 0, resulting in where P out is the population of the output state after the gate operation and is the Lewis-Riesenfeld phase [99]. Then, the systematic error sensitivity can be defined as [102] q i := − 1 2 Substituting ϕ and β [see Eq. (19)] into Eq. (21), we obtain q i 0 [18,19,102], which means that the holonomic gates are insensitive to the systematic errors induced by the pulse imperfections.
The average fidelity of a gate over all possible initial states can be defined by [111,112] The evolution operator U , describing the actual dynamical evolution, is calculated with the total Hamiltonian H tot (t) = H 0 + H D (t). Using the above definition, in Fig. 3 the average fidelity is nearly 99.9%, indicating that our protocol is insensitive to the systematic error caused by pulse imperfections.
The average infidelities (1−F ) of arbitrary single-qubit gates are shown in Fig. 4(a). The left (right) side of each circle denotes the average infidelity in the absence (presence) of pulse imperfections. When considering pulse imperfections with an error coefficient δ i = 0.1, the infidelities only slightly increase from ∼ 10 −4 to ∼ 10 −3 . For instance, in the case of the Hadamard gate, pulse imperfections with an error coefficient δ i = 0.1 only increase the infidelity from < 10 −4 to ∼ 10 −3 . This indicates that the generated gates are mostly insensitive to systematic errors.
Generally, a geometric gate can be robust against noise caused by amplitude fluctuations. Without loss of generality, we use additive white Gaussian noise to investigate the influence of such noise. In this case, the driving amplitudes Ω k should be corrected to be Ω s k = AWGN(Ω k , r). Here, AWGN(Ω k , r) is a function that generates the additive white Gaussian noise (AWGN) to the original signal Ω k with a signal-to-noise ratio r. Because the additive white Gaussian noise is generated randomly in each single simulation, we perform the numerical simulation 20 times to estimate its average influence [see Fig. 4(b) with an illustraton of the Hadamard gate]. As shown in Fig. 4(b), when considering relatively strong noises with r = 15, the gate fidelities can still be higher than 99%. This indicates that our protocol is mostly insensitive to noise caused by amplitude fluctuations.
To check the robustness of the geometric gates against decoherence, we assume the input state as |ψ in = |0 , corresponding to an output state |ψ out = U T |ψ in . Using (Θ s , θ, φ) = (π/2, π/4, 0) (Hadamard gate), in Fig. 5(a) we show the fidelity F out = ψ out |ρ(t f )|ψ out versus γ and κ in the presence of pulse imperfections when δ i = 0.1. In this figure we notice that the dissipation and dephasing of the atom affect the evolution much weaker than those of the cavity. For experimentally realistic parameters of superconducting circuit experiments [55], 0.33, 0.3, 8, 8) kHz, the fidelity of the output state is F out 99.56%, indicating that our protocol is robust against decoherence.

Hamiltonian is described by
The eigenstates of H R corresponding to the eigenvalues E m can be described by where |n a b and |n b b denote the Fock states of the two cavity modes, respectively. Then, we assume that the driving field is When choosing the frequencies that ω a /ω b = 0, 1, 2 . . ., and the effective Hamiltonian is approximatively For simplicity, we assume that the intermediate state is the dressed state |ζ 0 , the driving amplitudes become and the phases are Here, the auxiliary parameter φ00 is time-dependent and the auxiliary parameter φ is time-independent. The effective Hamiltonian in Eq. (28) becomes with the binomial codes Here, the bright state |b can be defined as For simplicity, we choose θ 0, (1,2) to be time-independent. The orthogonal partners of the state |b become Then, by using the same strategy as that of the single-qubit case, we choose Ξ 0 (t) sin φ00 =Ω p (β, ϕ)/2 =β cot ϕ sin β +φ cos β, Ξ 0 (t) cos φ00 =Ω s (β , ϕ)/2 =β cot ϕ cos β −φ sin β.
The evolution operator after a cyclic evolution along |ψ − (t) = ie iβ (ϕ/2)|µ |b + sin(ϕ/2)|ζ 0 , in the subspace spanned by {|b , |d 1 , |d 2 , |d 3 } is given by In the computational subspace spanned by These two-qubit gates using the same strategy as the single-qubit case are also insensitive to the errors induced by pulse imperfections. Therefore, when considering the error coefficient δ i = 0.1, in Fig. 5(b), we show that arbitrary two-qubit gates can be implemented with high fidelities.
In the presence of decoherence, the populations [red-solid broken line in Fig. 6(a)], calculated using the master equation in Eq. (23) are almost the same as those calculated using the coherent dynamics when feasible parameters are considered. This indicates that our protocol for the state preparations is robust against decoherence.

VII. CONCLUSION
We have investigated the possibility of using USC systems for the implementation of fast, robust, and fault-tolerant holonomic computation. The dressed-state properties of the USC systems allow to simultaneously couple the dressed state |ζ m to multiple Fock sates, such that one can manipulate the population and the phase of each Fock state as desired. The binomial codes formed from these Fock states are protected against the single-photon loss, making the computation fault-tolerant. Moreover, by designing the pulses with invariant-based engineering, we can eliminate the dynamical phase and achieve only the geometric phase in a cyclic evolution. Such a control technique is compatible with the systematic-error-sensitivity nullification method, making the evolution mostly insensitive to the systematic errors caused by pulse imperfections. Additionally, using the USC regime allows to apply relatively strong driving fields, such that our protocols are fast. As results, our protocols are robust against the decays and dephasings of the cavity and the atom. Note that this work can freely control a bosonic mode. The proposed idea can be generalized to realize NHQC with other bosonic error-correction qubits, such as cat-qubits [48,51], for fast, robust, and fault-tolerant quantum computation.
Some experimental observations of the ultrastrong light-matter coupling in superconducting quantum circuits are listed in Table II. To reach the ultrastrong and deep-strong coupling regimes, we can choose a setup with a flux qubit coupled to a lumped-element LC resonator [104,135]. In such superconducting circuit experiments, qubit and resonator frequencies are usually in the range ω c,(q) /2π ∼ 1-10 GHz. Thus, we choose g/ω c 0.7 (0.8) and ω c /2π = 6.25 GHz, which are experimentally feasible, as shown in Table II.
Recent experimental work has demonstrated that dissipation and dephasing rates in a flux qubit is of the order of 2π × 10 kHz [73,137,138]. The transmon qubits, which have lower anharmonicity than flux qubits, can have dissipation and dephasing rates approaching 2π × 1 kHz [139,140]. For transmission-line resonators, quality factors factors Q = ω c /κ on the order of 10 6 have been realized [141], which indicates that quantum coherence of single photons up to 1 ∼ 10 ms is within current experimental capabilities [142]. Therefore, our proposal works well in the USC regime, and it may find compelling applications for quantum information processing for various USC systems, in particular, superconducting systems. The total Hamiltonian for this protocol can be written as (ω µ + nω c )|µ µ| ⊗ |n n|, H D (t) = Ω(|µ g| + |g µ|). (A1) Here, |ζ m are the dressed eigenstates of the Rabi Hamiltonian with eigenvalues E m , ω µ denotes the energy of the lowest atomic level |µ , n is the cavity photon number, and Ω = Ω k cos(ω k t + φ k ) is a composite pulse driving the atomic transition |µ ↔ |g . Performing the unitary transformation U d = exp(−iH 0 t/ ) and choosing the frequencies as ω k = E m − ω µ − kω c , we have where ∆E m,m = E m − E m is the energy gap between the eigenstates |ζ m and |ζ m . Obviously, when satisfying the fast-oscillating terms can be neglected in the rotating wave approximation (RWA). Then, the effective Hamiltonian becomes i.e., the effective Hamiltonian in Eq. (6). The coefficients c m n = ζ m |g |n and d m n±1 = ζ m |e |n ± 1 can be obtained numerically [see Fig. 7(a) as an example for the ground dressed state |ζ 0 ]. According to our numerical results, when 0.5 g/ω c 1, the probability amplitudes c 2 0 , c 2 2 , c 2 4 of the states (|g |0 , |g |2 , |g |4 ) in the third dressed state |ζ 2 are greater [see Fig. 7(b)] than in the others. Thus, when focusing on manipulating the Fock states (|0 , |2 , |4 ), the effective driving intensities (i.e., c 2 0 Ω 0 , c 2 2 Ω 2 , and c 2 4 Ω 4 ) can be much stronger by using |ζ 2 to be the intermediate state. Therefore, the gate time can be shortened. In Fig. 7(b), we find that the coefficients c m n jump from zero to nonzero values when g/ω c 0.43. This is caused by an avoided level crossing [72,73] when g/ω c 0.43 [see the red circle in Fig. 7(c)]. In Fig. 7(c) we notice that the dressed states become nonequidistant as the energy gap ∆E m,m+1 = constant when 0.1 g/ω c 1. For instance, when g/ω c ∼ 0.5, we have |∆E m,m+1 − ∆E m+1,m+2 | 0.5ω c . This indicates that the USC can induce strong anharmonicity in the dressed states |ζ m .

Appendix B: Dynamical and geometric phases
An operator I(t) satisfying ∂ t I(t) = i[H(t), I(t)] is a dynamical invariant of an arbitrary Hamiltonian H(t). According to [99], an arbitrary solution of the Schrödinger equation can be expressed by using the eigenstates of I(t) as |ψ(t) = n C n e iRn(t) |ψ n (t) , where C n are time-independent amplitudes, |ψ n (t) are the orthonormal eigenvectors of I(t), and R n (t) are the Lewis-Riesenfeld phases [99]. These phases include dynamical phases ϑ n (t) = − 1 t 0 ψ n (t )|H(t )|ψ n (t ) dt , and geometric phases Θ n (t) = t 0 ψ n (t )|i∂ t |ψ n (t ) dt .
For instance, when ψ(0)|ψ 0 (0) = 1, we have C 0 = 1 and C n =0 = 0. The evolution of the system is exactly along the eigenstate |ψ 0 (t) , which is a shortcut to the adiabatic passage of H(t).