A Rout to Protect Quantum Gates constructed via quantum walks from Noises

The continuous-time quantum walk on a one-dimensional graph of odd number of sites with an on-site potential at the center is studied. We show that such a quantum-walk system can construct an X-gate of a single qubit as well as a control gate for two qubits, when the potential is much larger than the hopping strength. We investigate the decoherence effect and find that the coherence time can be enhanced by either increasing the number of sites on the graph or the ratio of the potential to the hopping strength, which is expected to motivate the design of the quantum gate with long coherence time. We also suggest several experimental proposals to realize such a system.


APPENDIX A: DETAILS OF THE EFFECTIVE HAMILTONIAN OF THE COUPLED LC CIRCUITS
With the help of the Kirchhoff's law, the motion equations of particles for the left series of the superconductor LC circuits can be given as where L 1(2) refers to the self-inductance, M to the mutual-inductance, C to the capacitance, and V to the AC voltage provided by the microwave resonator. According to the above motion equations of particles, we can write the Lagrangian as With the help of canonical equation, we can obtain a set of canonical momentum Then the classical Hamiltonian describing the left series of the superconductor circuits is where In order to quantized the classical Hamiltonian (A1), we introduce two creation and annihilation operators a j = 1 ) with (j = 1, 2) sincep j andq j satisfy the canonical communication relation [q j ,p j ′ ] = iδ jj ′ . Then the two-order quantization form of the classical Hamiltonian (A1) iŝ where ω j = 1 √ µ j C . If we take the coefficients to satisfy the condition µ 1 = µ 2 = µ and λ 1 λ 2 = C 2 µ 1 µ 2 through tuning L 1(2) , M and C, the Hamiltonian of the left series of the superconductor LC circuits can be simplified aŝ where ω = 1 √ µC , J = 1 µ and the constant term is neglected without affecting the physical properties of the system.
The Hamiltonian describing the right series of the superconductor LC circuits is in the same form withĤ L just replacing a 1(2) by a 3(4) , respectively, The microwave resonator bridging the two series of superconductor LC circuits can be quantized just like the LC circuit. For a particular mode k = 2π/d, the voltage in the microwave resonator in Schödinger picture iŝ where ω 0 = 2π/(d √ L 0 C 0 ) with d referring to the length of the resonator, L 0 and C 0 to the inductance and capacitance per unit length of the resonator, respectively. Then the AC voltage V in Eq. (A2) and Eq. (A3) can be quantized with the help of So the coupling term between the resonator and the superconductor LC circuits, i.e., the sum of the third term in Eq. (A2) and Eq. (A3), can be written aŝ In rotating-wave approximation, the coupling between the resonator and the superconductor LC circuits can be rewritten asĤ Taking λ = J via tuning the parameter of the system, we have the total Hamiltonian of the superconductor LC circuits bridged by a microwave resonator iŝ Considering a single-photon process, the Hilbert space of the system describing by Eq. (A4) is spanned by the following vector, where |vac⟩ denotes a vacuum state. Then the Hamiltonian Eq. (A4) can be expressed as where E = ω 0 − ω and the constant term is neglected. That is just the Hamiltonian describing the continuous-time quantum walk of a single particle on the graph of five sites we considered.

APPENDIX B: CALCULATION OF SEVERAL NOISES WITH DIFFERENT SPECTRA
Considering several noises with different spectra, we calculate the corresponding characteristic time of coherence τ and compare it with the operation time τ 0 . τ 0 = (n + 1)Ē J 2 π 2λ(m) , which implies that τ 0 ∝ κ with κ = (n + 1)Ē J .

Constant Spectrum
The spectrum of the correlation function is For the case of large ω 0 , which implies that τ ∝ κ 2 . When the ratioĒ/J or the length of the chain increases, τ increases faster than τ 0 .

The corresponding I(t) is
which shows that τ ∝ κ 2 .

Gaussian Spectrum
The spectrum of the correlation function is The corresponding I(t) is which shows that I(t) monotonically increases with t. Thus we evaluate a characteristic time τ by using following equation Since I(t) is monotonically increasing and I(0) = 0, τ is also very large for large γ. In this regime, the equation (B2) can be simplified as Then we obtain following relation which implies that τ (γ) is proportional to γ. So we have τ ∝ κ 2 .

1/f Spectrum
The spectrum of the correlation function is For the given lower and higher cutoffs ω l , ω h , we have where the function f (t) can be expressed as The functional image for f (t) is plotted in Fig. 1. One can see from the functional image that −1 < f (t) < 0 and lim t→+∞ f (t) = − 1 4 .
For the short-time limit ω h t ≪ 1, one can find that I(t) ∝ t 2 . Whereas, for the long-time limit ω l t ≫ 1, one can see . These analytical results imply that increasing the chain's length can protect the system from 1/f noises effectively in the case of ω l τ n ≫ 1, where τ n is operation time.