Quantum PICs: Empowering designers with accurate simulations
Integrated photonics is an exciting potential route to quantum computation and communication. The Ansys Lumerical qINTERCONNECT solver can precisely simulate quantum PICs, enabling engineers to design these circuits without needing extensive quantum mechanics knowledge.
By Sebastian Gitt, Ruoshi Xu, and Ahsan Alam, R&D engineers at Ansys
Of the many promises of the quantum revolution, two of the most exciting are powerful quantum computing and secure quantum communication channels impervious to tampering. Researchers are making rapid progress in exploring integrated photonics as a potential route to both of these. However, crafting photonic circuits for these applications often demands a profound grasp of quantum mechanics and the nuanced behaviour of qubits – the elemental units of quantum information.
To expedite the advancement of this technology, there is therefore a pressing industry demand for a commercially accessible tool that can empower engineers who have limited quantum mechanics expertise to design and refine these circuits.Furthermore, as these quantum PICs grow larger and more sophisticated, it is increasingly critical to suppress errors due to circuit imperfections, which impede their reliability and scalability.
These flaws include manufacturing variances in circuit components, photons that are not perfectly indistinguishable, and losses in the circuit. Consequently, there is a growing need for simulation tools that can also model realistic quantum PICs.
For purely classical photonic chips, the simulation can be performed by commercially available tools like Ansys Lumerical INTERCONNECT, which can accurately model the evolution of a single photon in any PIC by calculating the S-parameter. However, for scenarios with more than one photon, non-classical interference between photons must be accounted for.
Figure 1. A circuit for a user-defined unitary matrix of any size can be constructed using Ansys Lumerical qINTERCONNECT. Using the Clements decomposition method, the circuit consists of a square mesh of beamsplitters, which in turn can be customised by the user to contain custom components or to be a different geometry altogether.
Enter the Lumerical qINTERCONNECT solver by Ansys, a cutting-edge solution poised to address this need. By facilitating precise simulation of quantum PICs, compatible with Process Design Kits (PDKs) from photonic foundries, qINTERCONNECT streamlines the design and simulation process for engineers, supporting them to navigate this intricate domain with confidence and ease.
The qINTERCONNECT solver uses the classical scattering matrix from INTERCONNECT to generate a quantum scattering matrix, which can then be used to calculate the evolution of a quantum state in a PIC [1]. By using the quantum state in the Fock (number) basis as input to the solver, qINTERCONNECT returns the output of the PIC as a density matrix. Additionally, the solver can report the fidelity of the final state with respect to an expected final state and the probability of success for measuring a given output.
Since qINTERCONNECT uses INTERCONNECT as a starting point to simulate the PIC, it benefits from the advanced features of INTERCONNECT. These include the ability to design with PDKs from different photonic foundries, to account for manufacturing variability in photonic components by using statistically enabled compact models, and advanced nonlinear model support for elements like nonlinear optical fibres.
While qINTERCONNECT works in the Fock basis, it supports more exotic inputs in terms of Fock states. Users can encode logical qubit states in terms of physical Fock states, as well as coherent states and squeezed states. Additionally, the output from non-classical photon sources, such as physics-based simulations of optical nonlinearities, can be used as inputs.
Two areas that have recently attracted increased interest are programmable unitary matrices, which have uses across many different areas within quantum photonics, and continuous-variable quantum key distribution (CV-QKD) for secure data communication. qINTERCONNECT, as we will demonstrate, can be used in both of these contexts.
Figure 2. (a) The probabilities of measuring up to three photons in the
first channel of the output of the unitary matrix circuit for an input
state of indistinguishable photons |1,1,1,0 >⟩ (red) and a state of
distinguishable photons (blue) at six different interaction strengths,
represented different times, as in [7]. (b) The probabilities of
measuring any given output state across all four channels at the output
of the unitary matrix circuit for the same indistinguishable state as in
(a). (c) Same as (b) but coupling gap values are taken from a normal
distribution with a mean value equal to the ideal coupling gap value of
200 nm and with a standard deviation of 10 nm.
Generating and simulating programmable unitary matrix circuits
Unitary operations are fundamental building blocks in both classical and quantum linear systems, since any linear operator can be decomposed into a product of unitary and diagonal operators [2]. In classical systems, they are commonly used in applications related to matrix multiplication and neural networks, whereas in quantum systems they can be used for boson sampling, graph similarity algorithms, and the simulation of quantum systems [3-5].
The main principle behind constructing a programmable unitary matrix lies in the fact that an N-dimensional unitary can be factorised into a product of 2-dimensional unitary matrices [6]. Hence, a collection of programmable 2-dimensional unitary matrices can be used to construct a programmable unitary matrix of any size.
Since Mach-Zehnder interferometers (MZIs) can be used for these 2-dimensional unitary matrices, integrated photonics is a particularly convenient platform for implementing programmable unitary matrices. A common implementation is the so-called Clements decomposition, which improves upon the design of the earlier proposed Reck decomposition by using a symmetrical design that reduces the optical depth of the circuit and is more robust to losses [7].
With Ansys Lumerical qINTERCONNECT, users can construct a PIC in INTERCONNECT representing their own user-defined unitary matrix of arbitrary size. Furthermore, custom circuit elements can be used in the construction of each MZI. This enables workflows where the designer starts with component design with Multiphysics tools like Ansys Lumerical FDTD, CHARGE, or MODE, then generates an accurate, statistically enabled compact model of the optimised element using CML Compiler, and finally creates the PIC in INTERCONNECT to simulate how quantum states will evolve through it.
Figure 3. An overview of the pilot-assisted CV-QKD circuit, from transmitter to receiver, being simulated in INTER-CONNECT.
Furthermore, the user has the option of using more complex MZIs as the fundamental building block in the unitary circuit, which can provide a greater robustness against manufacturing imperfections and losses, at the cost of increased circuit complexity [8]. Figure 1 depicts an overview of how the user constructs the unitary matrix circuit, as well as each hierarchical level of the circuit.
Once the INTERCONNECT project file containing the programmable unitary matrix circuit has been generated, it is ready for simulation in qINTERCONNECT, a quantum circuit solver that can calculate the frequency domain response of a PIC in the Fock, or photon number, basis. The solver takes the classical S-parameters given by INTERCONNECT and uses them to calculate a quantum S-matrix (QS-matrix), which relates input and output states defined in a Fock basis prescribed by the number of channels and a maximum total number of photons.
The QS-matrix specifies the quantum mechanical state of light exiting the circuit in terms of the non-classical input state. Consequently, the qINTERCONNECT solver enables users to specify an input state in the Fock basis, either as a state
in bra-ket notation or a density matrix, and the solver will yield the output state in terms of probability amplitudes for measuring discrete photon detection patterns in the output channels. Furthermore, if a target output state is supplied, the solver can also calculate the fidelity.
As an example, the qINTERCONNECT solver can be used to construct a photonic circuit to model the unitary evolution of a non-interacting Bose-Hubbard model, or “hopping” Hamiltonian with four sites. It can then simulate the probabilities of success for various measurement outcomes for three-photon input states of both distinguishable and indistinguishable photons.
A simulation was performed using an input state of |1,1,1,0>, corresponding to one photon in each of the first three channels and no photon in the final channel. Figure 2a and 2b show the measurement results for the output of the first channel using both distinguishable and indistinguishable photons, as well as the results for measuring all four channels using an input state consisting of indistinguishable photons, which agree with established results [5].
Using this workflow, it is possible to characterise the impact of manufacturing uncertainty on these measurement outcomes. For example, users can vary the coupler gap spacing in the directional coupler elements that make up each MZI in the unitary circuit, and then repeat the simulations, as illustrated in Figure 2c. Here, the coupling gap values are drawn from a distribution with a standard deviation of 10 nm, modelling the effects of fabrication variance. The impact this has on the measured probabilities of detecting various output states is clearly demonstrated.
Figure 4. (a) Calibration of the shot noise variance is accomplished by
measuring the shot noise from the detector when the signal is
disconnected and the LO is connected, and the electronic noise from when
both the signal and LO are disconnected. (b) The power spectra for the
shot noise and electronic noise.
Continuous-variable quantum key distribution
Ansys Lumerical tools can also be used to model circuits for continuous-variable quantum key distribution (CV-QKD). In these applications, keys are encoded in the quadrature components of strongly attenuated coherent states using quadrature phase shift keying (QPSK). Since the information in the quadrature components can be read out using homodyne detection, CV-QKD can be fully implemented using well-established telecom technology. This stands in contrast with discrete-variable QKD, which requires single-photon sources and detectors.
An eavesdropper intercepting the quantum signal will necessarily introduce additional noise into a CV-QKD system, which can be detected by the receiving party. If the noise level inherent in the system is minimised, the additional noise introduced due to the eavesdropper can be detected, and the protocol can be aborted until the channel is determined to be secure. Since the amount of noise tolerated in the system before the protocol is aborted is so low, it is important to be able to simulate the noise in the system accurately.
Given that the coherent states are weak, they cannot be used as an accurate phase reference to recover the encoded keys. However, a stronger pilot reference can be transmitted alongside the quantum signal for phase correction at the receiver side. To avoid crosstalk between the pilot and the signal, the pilot can be first shifted to a different frequency using suppressed carrier single-sideband modulation (SC-SSB), and then rotated to an orthogonal polarisation [9]. In addition, a pulse carver can be used to reduce the information that can be obtained from monitoring the transition between symbols. Figure 3 depicts an overview of the full CV-QKD circuit.
The most important figure of merit for a QKD system is the secure key rate, which, in general, is improved by minimising noise and losses in the system. Consequently, the goal of the simulation is to determine the transmittivity and total excess loss, the latter of which is attributable to the eavesdropper and must therefore be kept as low as possible.
Since the transmittivity and excess noise must be given in shot noise units, the first step is to perform a shot noise unit calibration, which consists of two simulations. First, the transmission line is disconnected from the receiver and a simulation is performed with only the local oscillator (LO) connected to the homodyne detectors to measure shot noise plus noise intrinsic to the detector. Next, a second simulation, with the LO disconnected, measures detector noise. The variances of both measured voltage values are subtracted to obtain the shot noise variance. These two steps are then repeated eight times to obtain an averaged value for the shot noise calibration. Figure 4 shows the power spectra for the shot noise and electrical background noise.
Figure 5. (a) Accumulated phase drift over time between the pilot and
the LO. (b) Digital signal processing (DSP) is performed using the
measured quadrature values of the pilot tone to correct to the phase of
the quantum signal. (c) Comparison of simulated parameters in
INTERCONNECT against experimentally determined values [9].
After the shot noise calibration is complete, a simulation is performed with the transmitter connected to the receiver and the measured quadrature values are normalised by the previously measured shot noise variance, which is necessary to relate the measured results to the language of Gaussian information [10]. As before, the results are averaged over eight simulations. Then, the pilot tone is used to correct the phase drift between the LO and the signal.
Next, a fraction of the key information is disclosed between the sender and the receiver, which allows the receiver to calculate the transmittivity and the total noise in the link. If the receiver is inaccessible to an eavesdropper, the receiver noise can, under a relaxed security assumption, be treated as trusted noise, and is no longer attributed to the eavesdropper. In this case, the total noise minus detector noise will be another important figure of merit (ξtot - ξdet). Comparing the simulated parameters to those obtained experimentally and reported in the literature, we can see consistent results in the transmittivity (T), signal-to-noise ratio (SNR), and total noise (ξtot), as shown in Figure 5.
Since the transmittivity is affected not only by the transmission channel losses but also by detector losses and any coupling losses between the transmission channel and the detector, its value will be strongly dependent on the details of any given experimental implementation. Then, assuming a reconciliation efficiency value of β = 0.97, a frame-error rate of FER = 0.05, and a fraction disclosed for parameter estimation of ν = 0.25, and using the parameters in Figure 5, the secret key rate is found to be 38.0 Mbit/s over 13 km [9].
A versatile tool for designing and refining quantum PICs
Thanks to Ansys Lumerical qINTERCONNECT, you no longer need to be an expert in quantum mechanics to design a functional and effective integrated quantum photonic circuit. By leveraging foundry PDKs and advanced statistical and nonlinear compact models for different building blocks in their PICs, designers can account for issues originating from fabrication variations, propagation losses, and indistinguishable photons.
Designers can investigate the evolution of quantum states propagating through their quantum PICs using various types of sources ranging from physical Fock states to logical qubit states, such as coherent states or squeezed states to even non-classical photon sources. This makes qINTERCONNECT a versatile tool for the design and evaluation of integrated quantum photonic circuits.
