† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11474049) and CAST Innovation Fund.
We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots. By analyzing the hybrid system, we obtain the necessary factors implementing a quantum walk in phase space: the walker, coin, coin flipping and conditional phase shift. The coin flipping is implemented by adding a driving field to the resonator. The interaction between the quantum dots and resonator is used to implement conditional phase shift. Furthermore, we show that with different driving fields the quantum walk in phase space exhibits a ballistic behavior over 25 steps and numerically analyze the factors influencing the spreading of the walker in phase space.
Quantum walk (QW) is appealing as an intuitive model in quantum algorithms[2–4] and quantum simulations,[5–8] because it exponentially speeds up the hitting time in glued tree graphs. Furthermore, QW offers a quadratic gain over classical algorithms on account of the diffusion spread (standard deviation), which is proportional to elapsed time t, rather than
Recently, quantum computing with quantum dots has made huge progress,[31–38,40–42] and the technique for coupling electrons associated with a semiconductor double-dot molecule to a microwave stripline resonator has become more and more matured. Here we make use of this technology and propose the implementation of a one-dimensional QW in phase space (PS) with superconducting resonator-assisted quantum double-dot. The walker is presented by a coplanar transmission line resonator with a single mode, and a two-level system — one electron shared by double dots via tunneling serves as the quantum coin.
In our scheme the QW is executed with indirect flipping of the coin via directly driving the resonator and allows controllable decoherence over circles in PS for observing the transition between QW and RW.[11,39,43] In the next section, we give a brief introduction of the QW in PS. In Section 3, we implement QW via realizing the walker, coin, coin flipping and conditional phase shift. In addition to the numerical analysis under the different driving fields, we observe the ballistic behavior of QW in PS and the QW–RW transmission with the influence of decoherence introduced by the shift operation in the position space.
Like for the QW on a line in position space in which the walker moves towards the left or right based on the coin state, for QW on a circle in PS, the walker rotates either clockwise or counter-clockwise along the circle in PS by the same amount, say an angle Δθ, with the strictly random choice of ±Δθ through the impulse, which is applied by a harmonic oscillator.
In an ideal QW on a circle, the coin is replaced by a two-level system with internal states |0〉 and |1〉. Here we introduce the finite-dimensional orthogonal phase state representation
where |n〉 is the Fock state. If the step size Δθ = 2π/d, d ∈ N, then the walker always remains on the circle with angular lattice spacing Δθ. The walker walks in PS with a state |θk〉 which can be decomposed into the phase states. Introduce the rotation operator R̂m = einθm, m ∈ Z, and we will have R̂m|θk〉 = |θk+m〉. We choose the Hadamard operator
as the coin flipping operator and the unitary operation of one step of the QW in PS is given by
with the number operator on the walker state n = a†a, a† and a are the creation and annihilation operators, respectively, and
We define the initial state of walker+coin system as
where |φ0〉 is the initial coin state. After N steps, the system evolves into
The walker’s phase distribution on a circle is
with d being equally spaced values of θ = θk and ρw = Trc|ϕN〉〈ϕN| is the reduced density matrix of the walker after tracing out the coin.
The standard deviation of the phase distribution, σ, which is the symbol of the spreading of the QW, is linear with respect to time t. Therefore, in sufficiently short time, the relation of phase spreading with time on a circle is a power law and satisfies
where ζ is the slope coefficient and ξ is the intercept: ζ = 1 for the QW and ζ = 1/2 for the RW.
Circuit quantum electrodynamics (QED) is a device which is used to study the interaction between the quantum particle and the quantized electromagnetic mode inside a resonator. In this paper we consider a hybrid QED system of superconducting resonator-assisted quantum double-dot shown in Fig.
After applying the magnetic field to the quantum dots, the double-well potential forms a circuit. We just consider one circumstance, i.e., whether the electron is located in the left dot or right, the Hamiltonian describing the circuit is given by
where Ec = e2/2Ctot is the charge energy, Ctot = Cg + CJ is the total capacitance in the circuit, CJ is the Josephson capacitance and Cg is voltage biased from a lead having capacitance to the circuit, Ng = CgVg/2e is the gate charge which represents the total polarization charge. Restricting the gate charge to the range Ng ∈ [0,1] by using the voltage Vg, the Hamiltonian in Eq. (
So far, we have shown the Hamiltonian of the double-well potential. From Eq. (
Now we consider the circuit QED of double dots coupled to a superconducting resonator. The dots are located in the center of the resonator. If the oscillator mode of the resonator is coupled to the double-dot, by using the coordinate system transformation
and choosing 2Ng = 1, θ = π/2, the Hamiltonian of the interacting qubit and resonator system with the rotating wave approximation takes the form (ħ = 1)
where ωc is the frequency of the resonator, and Ω is the resonator-induced energy splitting of the qubit:
is the vacuum Rabi frequency. The form of Hamiltonian in Eq. (
In our hybrid system, the walker can be represented by the phase state of the single mode of the resonator and the coin is the two-level energy system. To implement the coin flipping operator, a microwave time-dependent driving field is applied to the circuit QED system with the form
where ωd is the frequency of driving field. Let ɛ(t) be a square wave, so ɛ is a constant when the field is turned on, while if it is zero, then the field is off. The Hamiltonian of the hybrid system, containing the driving field, is
In the dispersive regime,
To calculate the effective Hamiltonian from Eq. (
The free evolution
continues even when the driving field is off (ɛ = 0).
By choosing the coherent state |α = 3〉 and different values of ɛ, we show the probability distribution of the walker in PS at the 4th step with step size Δθ = 0.3 (Fig.
The evolution of the hybrid system is described by the effective Hamiltonian. The first term on the right-hand side in Eq. (
Now all the factors that the implementation of QW needs are fulfilled. To make the scheme work, the value of constant coefficient ɛ in the last term in Eq. (
We choose the initial coin state as
set the initial field in the resonator as coherent state
and obtain the realistic system parameters as follows:[50,51]
We show how a QW in PS can be implemented in a quantum quincunx created through superconducting resonator-assisted quantum double dots and how interpolation from a quantum to a random walk is implemented by controllable decoherence introduced by the displacement of the walker in position space. Our scheme shows how a QW with just one walker can be implemented in a realistic system. The coin flipping operation is implemented by driving the resonator directly, and at the same time the driving field also introduces the displacement of the walker in position space and pushes the walker off the circle in PS. Thus the displacement in position space is equivalent to decoherence on the walker in PS which is controlled by the strength of the driving field. With the strength of the driving field increasing the decoherence increases and we observe the QW–RW transition.
Although in our paper we make use of the decoherence introduced by the driving field to show the transition from QW to RW, which is one of the main points of our paper, for most of the applications of QW it requires quadratic enhancement of walker spreading. The decoherence induced by the driving field can be compensated for by the method in Ref. . The displacement of the walker in position space pushes the walker off the circle in PS by changing the mean photon number of the resonator field. Hence we can adjust the pulse duration each time ti according to the predicted mean photon number n̄(i), that is, ti = [δ + 2n̄(i)−2gɛ/δ]π/4gɛ, to compensate for the effect due to the displacement and obtain a perfect QW in PS.