SUPPLEMENTARY INFORMATION FOR CHAPTER 5 - DEVICE CHARACTERIZATION, MODELING, AND CONTROL
E.4 Theoretical Analysis of Qubit-Photon “Gate" Errors
The cluster state generation protocol can be theoretically described by the quantum circuit illustrated in Figure F.5. This quantum circuit consists of single-qubit gates for controlling 𝑄𝐸 and qubit-photon gates; where the qubit-photon gates are the CNOT gate and the 𝐶 𝑍 gate. While in practice the “CNOT gate" consists of excitation from |𝑒⟩ → |𝑓⟩ by a single qubit gate and emission from |𝑓⟩, thereby placing a photon in the SLWG, it can still be instructive to consider this process as effectively another “gate."
Following a similar analysis to Ref. [60], we calculate the fidelities of these qubit- photon gates based on experimentally relevant parameters. For the effective CNOT gate (assuming a perfect |𝑒⟩ → |𝑓⟩ rotation), the main source of infidelity will be residual population in the|𝑓⟩state as discussed in Appendix E.2; which essentially can be regarded as a leakage error where the qubit leaves its computational subspace.
For the𝐶 𝑍gate, the main source of infidelity will be the wavepacket distortion of the photon after it scatters on𝑄𝐸 (see Chapter 2.2 for discussion of this phenomenon), as well a scattering phase different than𝜋. The wavepacket distortion will lead to the mode shape of the photon to differ from its true mode-matching function 𝑓(𝑡)when the qubit is in the |𝑒⟩ state. This results in a mode-matching inefficiency that can be regarded as a leakage error where thephotonleaves its computational subspace;
i.e., the subspace spanned by vacuum state and the state with an excitation in the wavepacket mode of interest with shape 𝑓(𝑡).
We can calculate the average fidelity of gates that suffer from such leakage error via the formula derived in Ref. [224], where “average fidelity" here corresponds to assessing the “error" of the gate uniformly over all pure initial states in the system Hilbert space. The formula derived is given by
𝐹 = 1 𝑑2+𝑑
h Tr
𝑀comp𝑀†
comp
+
Tr 𝑀comp
2i
, (E.8)
where𝑀comp = 𝑃𝑈†
0𝑈 𝑃, where𝑈0is the target gate and𝑈is the effective gate and
CZ F idelit y
0.95
0.997 0.97
0.98 0.99 1
0.998 0.999 1
0.96
0.0125 0.01 0.075 0.05 0.025
0 10 20 30 40 50
Residual Population
Γ
1D/σ
CNO T F idelit y
Figure E.7: Theoretical𝐶 𝑍 and CNOT Qubit-Photon Gate Fidelities. Theoretically calculated qubit-photon CNOT fidelities as a function of residual|𝑓⟩state population, and theoretically calcu- lated qubit-photon𝐶 𝑍fidelities as a function of Γ1D/𝜎, where𝜎 is the bandwidth of a Gaussian wavepacket; see Appendix text for more details. The black dashed line indicates the realized values of residual population and Γ1D/𝜎 in our experiment. For the dashed blue curve, the Gaussian wavepacket is truncated outside of 2𝜎, where for the solid blue curve the Gaussian wavepacket is truncated outside of 4𝜎. In our experiment, our emission protocol effectively truncated the Gaussian pulse outside of 2𝜎.
these are defined on the full Hilbert space of the systemincluding the leakage states, and𝑃is the projection operator onto the computational subspace. Note that𝑀comp will not be unitary if there is leakage out of the computational subspace, and this property will thus lead to a computed infidelity through Equation E.8.
For the CNOT gate, the |𝑓⟩ state residual population directly gives the magnitude of the leakage. For the CZ gate, the leakage and phase error may be computed by calculating the mode-matching integral∫
𝑑 𝑡 𝑓∗(𝑡)𝑔(𝑡), where 𝑔(𝑡) is the resultant distorted wavepacket when the 𝑄𝐸 is in the |𝑒⟩ state. Defining the frequency dependent scattering response of 𝑄𝐸 as𝑟(𝜔), we may write in the Fourier basis 𝑔(𝑡) =∫
𝑑 𝜔 𝑓(𝜔)𝑟(𝜔)𝑒−𝑖𝜔𝑡, where 𝑓(𝜔)is the Fourier Transform of 𝑓(𝑡). Further, we note that since the SLWG dispersion is approximately linear in the middle of the passband and our qubit is “end-coupled" to the waveguide, the scattering response 𝑟(𝜔) will be approximately given by Equation 2.37. Further, given that Γ1D ≫ Γ′ in our system, we may assume that𝑟(𝜔) = 𝑒𝑖 𝜃𝑟(𝜔), where the magnitude of 𝑄𝐸’s
scattering response is unit-valued for all frequencies. By calculating the overlap integral in the Fourier basis, we arrive at the following result:
∫
𝑑 𝑡 𝑓∗(𝑡)𝑔(𝑡)=
∫
𝑑 𝜔|𝑓(𝜔) |2𝑒𝑖 𝜃𝑟(𝜔), (E.9) where it is evident that the overlap integral is simply the “weighted average" of the scattering phase across the wavepacket’s power spectrum. Thus, if the wavepacket’s bandwidth is significantly smaller than the “width" of 𝑟(𝜔), which is set by Γ1D, then this weighted average of the phase will be approximately equal to -1.
In Figure E.7 we plot fidelities for the𝐶 𝑍 and CNOT gates that were numerically calculated via Equation E.8. For the CNOT gate we calculate the fidelity as a function of residual population, where for the purposes of this calculation the following CNOT matrix element is given by ⟨11|CNOT|10⟩ = √︁
1− 𝑝res, where 𝑝resis the residual population (and all other matrix elements are standard). Further, for the CZ gate we assume a Gaussian wavepacket with width 𝜎, we assume that 𝑟(𝜔)is given by Equation 2.37, and we calculate the fidelity as a function ofΓ1D/𝜎, where the 𝐶 𝑍 matrix element ⟨11|𝐶 𝑍|10⟩ = ∫
𝑑 𝑡 𝑓∗(𝑡)𝑔(𝑡) (and all other matrix elements are standard). We indicate with a dotted line the expected theoretical gate fidelities given our experimental parameters of Γ𝑒 𝑓
1D/𝜎 ∼ 1/14 and 𝑝res ∼ 0.01.
Moreover, for the𝐶 𝑍 fidelity we perform the calculation for a Gaussian pulse that is truncated 2𝜎away from the mean, and for a Gaussian pulse that is truncated 5𝜎 away from the mean, where outside the truncation window the pulse amplitude is set to 0.
From the plotted curves, it is evident that the CZ gate fidelity has a non-linear dependence onΓ1D, while the CNOT gate has a linear dependence on the residual population, in agreement with Ref. [60]. Further, we note that the infidelities calculated here in general are smaller than the magnitude of leakage, because leakage due to the gate only happens forsomeof the states of the computational subspace (for example, no CZ leakage error occurs if the photon is in the |0⟩ state). Finally, we point out to the reader that although our theoretically expected𝐶 𝑍gate fidelities are only 97% given our experimental parameters, by merely increasing Γ1D by a factor of 3 and minimizing spurious effects of pulse truncation, we can expect a𝐶 𝑍 gate infidelity∼ 10−4.
A p p e n d i x F