Silicon photonic processor of two-qubit entangling quantum logic

The strong nonlinear properties of silicon waveguides are well known [47]. SFWM, an effect of the ${\chi }^{(3)}$ nonlinearity, is now commonly used to produce photon pairs in silicon quantum photonic devices [21, 28].

In the non-degenerate SFWM process used here, two photons from a bright pump are annihilated, producing two correlated photons with different wavelengths (figure 1(a)). The two generated photons, 'signal' and 'idler', emerge spectrally on either side of the pump, conserving energy and momentum. In our experiment, spiralled 21 mm long waveguides were used to produce photon-pairs, with the pump, signal, and idler photon wavelengths being 1551, 1547 and 1555 nm. These photons were generated in a continuous spectrum and the chosen wavelengths were post-selected by the off-chip demultiplexers.

Our device uses a new scheme to generate entangled path-encoded states, which can subsequently be interfered, using pairs of non-degenerate photons. Pump laser is distributed between two reverse-HOM structures using a reconfigurable power splitter (splitting ratio ${\sin }^{2}[{\phi }_{\beta }/2]$). Each RHOM contains two spiralled waveguides and a thermal phase shifter, as in [28]. The internal RHOM phases (${\phi }_{T}$ and ${\phi }_{B}$) were set to $\pi /2$, such that the produced photon-pairs emerged deterministically split, one in each output waveguide, and in a state symmetrical between signal and idler photons. ${\phi }_{\beta }$ allows us to control the balance of photon-pair emission between the two RHOM structures, and so to control the entanglement present in the two-qubit output state.

Following figure 1(b), if ${\phi }_{\beta }=\pi$, photons will be generated only in the top RHOM, and the photon number output state, after the waveguide crossing, will be $| {\rm{\text{1010}}}\rangle$, or $| 00\rangle$ in the qubit basis. On the other hand, if ${\phi }_{\beta }=0$, only the bottom RHOM generates photons, leading to $| {\rm{\text{0101}}}\rangle =| 11\rangle$. Finally, if ${\phi }_{\beta }=\pi /2$, we obtain the maximally entangled state: $| {{\rm{\Phi }}}^{{\rm{\Theta }}}\rangle \equiv (| 00\rangle +{{\rm{e}}}^{{\rm{i}}{\rm{\Theta }}}| 11\rangle )/\sqrt{2}$, where Θ is a fixed phase factor due to the chip's intrinsic path-length mismatch. Thus, the output state from the entangled qubit generator is

$$\begin{eqnarray}&&| \psi \rangle =\sqrt{\beta }| 00\rangle +{{\rm{e}}}^{{\rm{i}}{\rm{\Theta }}}\sqrt{1-\beta }| 11\rangle \end{eqnarray} \tag{ 1 }$$

which can be continuously varied across a wide range of separable and entangled states, depending on the balance parameter, β. The balance depends on the square of the power division of the state control Mach–Zehnder interferometers (MZI) (controlled by the phase ${\phi }_{\beta }$), due to the two-photon dependence of SFWM:

$$\begin{eqnarray}&&\beta ={\left|\displaystyle \frac{{\sin }^{2}({\phi }_{\beta }/2)}{\sqrt{{\sin }^{4}({\phi }_{\beta }/2)+{\cos }^{4}({\phi }_{\beta }/2)}}\right|}^{2}.\end{eqnarray} \tag{ 2 }$$

The state $| \psi \rangle$ is fed into a two-qubit circuit, composed of single-qubit rotations, and a switchable entangling gate. We implemented the arbitrary rotations on each qubit by cascading phase-shifters and MZI. These were used to realise ${\hat{R}}_{z}$ and ${\hat{R}}_{y}$ rotations, respectively, obtaining an arbitrary $\mathrm{SU}(2)$ with the combination ${\hat{R}}_{z}\cdot {\hat{R}}_{y}\cdot {\hat{R}}_{z}$.

We implemented a switchable entangling gate using a scheme based on [44], but replacing the 1/3 beam-splitters with tunable-reflectivity MZIs. In this way, we can switch the gate's controlled-Z operation on and off. When on, the ${\rm{cz}}$ operation succeeds with probability 1/9. In the remaining 8/9 cases, non-qubit states are generated, which are filtered by the coincidence-counting post-selection. Note that only the on ($\cos ({\theta }_{{\rm{cz}}})=1/3$) and off ($\cos ({\theta }_{{\rm{cz}}})=-1$) gate configurations produce unitary operations. The two qubit gate is followed by rotations (parametrised by ${\theta }_{{Mz}3}$, ${\theta }_{{My}2}$, $M\in \{T,B\}$) used to implement quantum state tomography, via the method described in [48].

Since the phase shifter parameters (phase-per-electrical-power, and phase offset) varied between phase modulators, a calibration process was essential. Measuring the bright-light transmission from the inputs ($\mathrm{IN}$ and $\mathrm{IN}^{\prime}$) to the outputs (${\mathrm{OUT}}_{T}$, ${\mathrm{OUT}}_{B}$, ${{\mathrm{OUT}}_{T}^{{\prime} }}_{}$, ${{\mathrm{OUT}}_{B}^{{\prime} }}_{}$), we were able to characterise the electro-optic parameters of each thermal phase shifter, in a similar way to that described in [49]. We learned the parameters associated with each phase according to the scheme:

$$\begin{eqnarray}\mathrm{IN}^{\prime} & \to & {{\mathrm{OUT}}_{T}^{{\prime} }}_{},{{{\mathrm{OUT}}_{B}^{{\prime} }}_{}^{}}_{}\,:{\phi }_{B},{\theta }_{{By}1},{\theta }_{\mathrm{CZB}},{\theta }_{{Ty}1},{\theta }_{\mathrm{CZT}}\\ \mathrm{IN} & \to & {{\mathrm{OUT}}_{T}^{{\prime} }}_{},{{{\mathrm{OUT}}_{B}^{{\prime} }}_{}^{}}_{}\,:{\phi }_{\beta },{\phi }_{T},{\theta }_{{Tz}1},{\theta }_{{Bz}1}\\ \mathrm{IN} & \to & {{\mathrm{OUT}}_{}^{}}_{T}\,:{\theta }_{\mathrm{CZC}},{\theta }_{{Ty}2},{\theta }_{{Tz}2},{\theta }_{{Tz}3}\\ \mathrm{IN} & \to & {{\mathrm{OUT}}_{}^{}}_{B}\,:{\theta }_{{By}2},{\theta }_{{Bz}2},{\theta }_{{Bz}3}.\end{eqnarray} \tag{ 3 }$$

We observed instabilities in the calibration data, due to changes in electrical contact resistance between our probe card and the on-chip gold pads. To mitigate this, we periodically recalibrated the on-chip parameters. Metallurgical wire-bonded contacts can prevent this in future. Low levels of thermal and common-ground crosstalk were observed but not compensated. Recent results suggest that crosstalk can be reduced through efficiency improvements, passive compensation methods, and by current driving of the thermal phase shifters [38, 49, 50].

The offsets of the tomographic z-rotation phases (${\theta }_{{Tz}3}$, ${\theta }_{{Bz}3}$) were left at zero, meaning that additional random (fixed) z rotations were applied to each qubit before measurement. This choice was necessitated by the combined difficulty of: (1) calibrating the nonlinear source phase with bright light, and (2) doing this for each setting of the gate, in the device's finite stability time.

One of the key metrics of a photon-pair source is its pair-generation efficiency [51]. This quantity is obtained from the photon-pair detection rate as a function of the input power, accounting for loss and detector efficiency. Inside the 1 nm wide signal and idler spectral bands, we measured a brightness of $20\,\mathrm{kHz}\,{\mathrm{mW}}^{-2}$.

The indistinguishability between photon-pair sources is also important. The contrast of the RHOM block's quantum interference fringes indicates the indistinguishability of the block's constituent photon-pair sources. We measured RHOM quantum interference fringes on each source by configuring the chip to maximise photon flux at the ${{\mathrm{OUT}}_{T}^{{\prime} }}_{}$ and ${{\mathrm{OUT}}_{B}^{{\prime} }}_{}$ outputs, then varying ${\phi }_{T}$ and ${\phi }_{B}$ to obtain the fringes of figure 2. We pumped the bottom source via the auxiliary input $\mathrm{IN}^{\prime}$, and the top source via $\mathrm{IN}$ and the state-control MZI, integrating each point for 5 s. We observed $C=93.2 \% \pm 1.4 \%$ and 72.9% ± 0.8% fringe contrasts, respectively, for the top and bottom sources. Here, $C=({N}_{\max }-{N}_{\min })/({N}_{\max }+{N}_{\min })$, where ${N}_{\max }$ and ${N}_{\min }$ are the accidental-subtracted maximum and minimum fitted count rates. The reduced contrasts can be explained by deviations (from the ideal $\eta =50 \%$) in the input evanescent couplers of each RHOM structure; they are compatible with reflectivity values of $\eta \approx 43 \%$ and $\eta \approx 36 \%$ for the top and bottom sources, respectively.

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We next quantified the device's control over entanglement. Quantum state tomography was used to extract the purity ($P=\mathrm{Tr}({\hat{\rho }}^{2})$ [41]), the CHSH parameter, a strict measurement of quantum correlations, and the Schmidt number, analogous to the number of pure states represented in a given density matrix. These last two metrics show how separable the state is. The CHSH inequality, $S(\hat{\rho })\leqslant 2$ [31, 42, 52], is violated when the state $\hat{\rho }$ cannot be represented by a local classical theory, indicating its entangled quantum nature. The Schmidt number, on the other hand, is an entanglement monotone and can give further evidence of the entangled or separable nature of $\hat{\rho }$ [42, 43, 53]. CHSH parameter values were obtained by computationally selecting an optimal measurement set for each of the states under analysis [31].

We analysed a wide set of separable and entangled quantum states produced by the two-qubit source. Fixing ${\phi }_{T}={\phi }_{B}=\pi /2$, we varied the phase of the state control MZI, ${\phi }_{\beta }$, between 0 and π to prepare variably entangled states in the form of (1). When $\beta =0$ or 1, separable states result, while when $\beta =1/2$, a maximally entangled state is produced. States obtained directly from the source (bypassing the gate) showed good agreement with (1). These were measured using the ${\mathrm{OUT}}_{T}^{\prime}$ and ${\mathrm{OUT}}_{B}^{\prime}$ auxiliary outputs (see figure 1(b)). Measured and calculated variations of the balance, Schmidt number, and CHSH parameter are plotted in figure 3, versus the state control parameter ${\phi }_{\beta }$.

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In figure 4 we show a sample of density matrices arising from the main device configurations, and we list their properties (purity, Schmidt number, CHSH parameter, and fidelity with the ideal z-rotated state) in table 1. Errors were obtained from Monte-Carlo simulations, based on 200 samples of Poissonian photon noise and accompanying tomographic reconstructions [54]. As expected, the $\hat{I}$-mode gate did not substantially affect the properties of the input states. The $\hat{{\rm{cz}}}$-mode gate, however, acted to entangle separable states, and separate entangled states, though it also degraded the purity. The limited contrast in the quantum interference of the two RHOM sources contributed to this reduction, by occasionally depositing two photons into one 'qubit'. Gate and tomography calibration errors likely also contributed.

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Table 1.  Purity, Schmidt number, CHSH parameters and fidelity for a variety of measured states. The Schmidt number and CHSH parameter indicate entanglement. $S\gt 2$ indicates the presence of non-local correlations [42], while K indicates the number of coefficients in the Schmidt decomposition of the state [43]. The fidelities $F^{\prime}$ reported are computed against the ideal state optimised over local R_z rotations, to compensate for the intrinsic random phase factor on each qubit.

Source state	Gate	Purity P	Schmidt number K	CHSH S	Fidelity $F^{\prime}$
$| 00\rangle$	Bypassed	0.995 ± 0.012	1.012 ± 0.011	1.577 ± 0.072	0.973 ± 0.011
$| 00\rangle$	$\hat{I}$	0.946 ± 0.031	1.034 ± 0.017	1.465 ± 0.064	0.962 ± 0.016
$| 11\rangle$	Bypassed	0.998 ± 0.008	1.004 ± 0.006	1.511 ± 0.049	0.984 ± 0.007
$| 11\rangle$	$\hat{I}$	0.949 ± 0.055	1.048 ± 0.037	1.601 ± 0.121	0.948 ± 0.031
$(| 00\rangle +| 11\rangle )/\sqrt{2}$	Bypassed	0.864 ± 0.019	1.905 ± 0.022	2.560 ± 0.037	0.909 ± 0.028
$(| 00\rangle +| 11\rangle )/\sqrt{2}$	$\hat{I}$	0.832 ± 0.040	1.936 ± 0.025	2.538 ± 0.072	0.900 ± 0.026
$| ++\rangle$	$\hat{{\rm{cz}}}$	0.931 ± 0.036	1.657 ± 0.045	2.560 ± 0.078	0.873 ± 0.038
$(| 00\rangle +| 11\rangle )/\sqrt{2}$	$\hat{{\rm{cz}}}$	0.900 ± 0.071	1.166 ± 0.055	1.907 ± 0.137	0.839 ± 0.013

Since the entangling gate operates on the input state's phase, we must examine with care the phase of the output state, $\arg [\hat{\rho }]$. The intrinsic and uncalibrated z-rotations on each qubit result in complicated phase pictures (figures 4(e) and (f)). To compare these to their ideal counterparts, we computationally applied ${\hat{R}}_{z}({\zeta }_{t})\otimes {\hat{R}}_{z}({\zeta }_{b})$ to the reconstructed output state, and optimised the fidelity over local z-rotations via ${\zeta }_{t}$ and ${\zeta }_{b}$. The resulting fidelities are listed in table 1 and the process is shown visually in figure 5.

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