6.2.1 Amplitude Damping Set-up
Using the dissipation gadget, the discretized gradient echo memory model used for LIQU i|⟩simulation is shown in Figure 6.6 below. The dissipation ancilla at the bottom is a qubit which is subjected to general amplitude damping. This is done by running idle gate at each Trotter step with the error probability of(1−exp(−γ∆t)). A LIQU i|⟩code
snapshot for the amplitude damping set-up is also shown in Figure 6.7.
The LIQU i|⟩simulation is performed on a Windows 7 desktop machine using Intel Core i7-4770 3.4 GHz (Haswell) with 16 GB of main memory. In the case of running 4-qubit gradient echo memory model (6 qubits in total), the runtime for 1,000 trajectories each with 10,000 Trotter steps is 3 hours.
6.2.2 Toy Example: Single-Atom Memory
Before delving into the discretized GEM simulation, we will test the LIQU i|⟩set-up in a much simpler situation, namely a single-atom memory. Here we consider a single two-level atom driven by a Gaussian continuous-mode single photon as given by Eq. (6.2). This problem has been investigated thoroughly by both conventional and quantum network approaches.
The coupling operator of the atom isL=√Γσ−. The atom is assumed to be free of
internal Hamiltonian dynamics and field scattering for simplicity, i.e.,H=0 andS=I. HereΓ>0 is the normalized coupling rate (Γ=1).
6.2 Models and Results 143 //Probability of amplitude damping
let probDamp_E = 1.0 - Math.Exp(-
gamma*dt)
let noise = Noise(circ,ket,models)
ket.TraceRun <- 0 noise.LogGates <- false noise.TraceWrap <- false noise.TraceNoise <- false noise.DampProb(0) <- 0.0 //single- photon ancilla noise.DampProb(1) <- 0.0 //qubit 1 noise.DampProb(2) <- 0.0 //qubit 2 ... noise.DampProb(Nq+1) <- probDamp_E //dissipation ancilla
Fig. 6.7 LIQU i|⟩code snapshot for the amplitude damping set-up.
By using Trotter decomposition technique and LIQU i|⟩, we were able to simulate the dynamics of this open quantum system. In our simulation, dissipation is induced by an ancilla which undergoes amplitude damping. In Figure 6.8, we plot the “detection time” histogram and the excited-state population for the case of Ω=1.5Γ, which is
known to be the optimal Gaussian wave packet for atomic excitation. The detection time is triggered by the state-collapse event of amplitude damping noise and histogram is plotted from 10,000 LIQU i|⟩runs.
This is consistent with theoretical analysis which shows that that the maximum excitation probability of an atom driven by a Gaussian-shaped single photon is about 80%[189]. Since we are essentially running a stochastic simulation on the input-output model, we have the benefit of being able to simulate the conditional dynamics and output observable statistics (photon detection in this case) if we choose to observe the output.
By sweeping the Gaussian bandwidth (largerΩmeans broader spectrum in frequency
domain and vice versa), we can get the maximum excitation (absorption) rate of our single atom model, as shown in Figure 6.9 (black curve). There is a very narrow range of bandwidth where the absorption peaks. The optical “depth” created by cascading
144 Chapter 6. Simulating Input-Output Quantum Systems with LIQU i|⟩ 0 5 10 15 20 Time 0.00 0.05 0.10 0.15 0.20 0.25 0.30 F requency
Single-photon pulse shape
0 5 10 15 20 Time 0.0 0.2 0.4 0.6 0.8 1.0 PE
Fig. 6.8 Histogram of photon detection events from a two-level atom driven by a Gaussian-shape single photon state. Inset: the excited state population of the atom. The photon wave-packet is included for reference (red curve in main and dotted curve in inset). Trotter step is chosen to be 2π×10−4, and the number of trajectories (realizations)
is 10,000.
absorbers of gradient echo memory can enhance the photon absorption as shown in the below section.
6.2.3 Discretized Gradient Echo Memory
Using the full GEM model with signal generating filter as in Eq. (6.7) and Figure 6.6, we can investigate the performance of the discrete memory in terms of photon absorption (efficiency). Due to computing resource and runtime constraints, only models of two- and four-atom GEM are considered in this report (4 and 6 qubits in total, respectively). Despite minimal system size, the performance enhancement is significant as shown in Figure 6.9.
Almost perfect absorption (100% excitation) can be achieved for some Gaussian wave packet shapes by using just 4 atoms in series. More importantly, a monotonic improvement over the full range of bandwidth is achieved by introducing more qubit resources. Notice that in the case of the multi-atom memory, the single excitation from
6.2 Models and Results 145
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Fig. 6.9 Single photon absorption probability as a function of photon bandwidth. The baseline (black) is the single atom model which has the max absorption of 80% for a very narrow range of bandwidth. The more atoms involve, the better absorption and also wider range of bandwidth can be achieved. Each data point is a LIQU i|⟩run of 1,000 trajectories. Solid line is polynomial fit.
the photon is distributed among all memory qubits in a Fourier-transformed (frequency) basis. Therefore, the choice of gradient is of importance and can be optimized for a given target input wave packet. Since we use a linear symmetric gradient and the number of atoms are even (thus missing a zero point), the absorption rate has bi-modal behavior, as is clearly seen in the four-atom case. If, as for continuous solid-state GEM, the gradient is a smooth continuous function, the memory can effectively capture the full Fourier spectrum of incoming photons.
Besides the fact that this discrete memory model can be implemented on a general purpose quantum computer, one major advantage is that we can potentially extend the range of gradient. Inhomogeneous broadening (by the electric field induced Stark shift, for instance) has difficulty in achieving the bandwidth required for telecommunication wavelength photon absorption. If implemented on a digital quantum computer, the gradient term is controlled by the Trotter decomposed unitary thus can be changed freely provided that the time step can be reduced for sufficient error bound.
146 Chapter 6. Simulating Input-Output Quantum Systems with LIQU i|⟩
This discretized memory model has close analogy to classical digital signal pro- cessing technology. The gradient quantum memory program running on the a quantum computer is equivalent to an Analog-to-Digital Converter (ADC), where (i) the Trotter step serves as the sampling rate; (ii) the resolution is dictated by the number of qubits; and (iii) the sampling method is frequency gradient mapping (classical ADC’s have a wide variety of sampling methods such as integrating and sigma-delta.)