Simulation of integrated photonic gates
Andrei-Emanuel Dragomir, Cristian George Ivan, Radu Ionicioiu
SSimulation of integrated photonic gates
Andrei-Emanuel Dragomir, Cristian George Ivan, and Radu Ionicioiu
Horia Hulubei National Institute of Physics and Nuclear Engineering, 077125 Bucharest–M˘agurele, Romania
Quantum technologies, such as quantum communication, sensing and imaging, need a platform which is flex-ible, miniaturizable and works at room temperature. Integrated photonics is a promising and fast-developingplatform. This requires to develop the right tools to design and fabricate arbitrary photonic quantum devices.Here we present an algorithm which, starting from a n -mode transformation U , designs a photonic device im-plementing U . Using this method we design integrated photonic devices which implement quantum gates withhigh fidelity. Apart from quantum computation, future applications include the design of photonic subroutinesor embedded quantum devices. These custom-designed photonic devices will implement in a single step a givenalgorithm and will be small, robust and fast compared to a fully-programmable processor. I. INTRODUCTION
Quantum information brings a paradigm shift of how werepresent, store, process and read information, with huge im-pact on future technology. Successful applications of quantumtechnologies include quantum communication/cryptography,quantum sensing, quantum simulation and quantum imaging.However, the ultimate goal is to design and build a quan-tum computer, i.e., a device which can implement any unitarytransformation U on an arbitrary quantum state.One of the approaches towards this objective is quantumoptics, where photons are the main carriers of quantum in-formation. Quantum algorithms are a set of transformations(gates) applied to qubits, the primary units of quantum infor-mation. In quantum optics, these gates are optical elements,such as beam-splitters, phase shifts, polarising beam-splittersetc. Most of the research in this field is done with bulk optics,i.e., macroscopic elements on an optical table (lenses, beam-splitters, wave-plates etc). However, the large size of thesecomponents prevents miniaturisation and scalability towardsmore complex quantum algorithms.A solution to this problem is integrated quantum photonics[1]. The goal is to implement on-chip every part of a quan-tum circuit: qubit generation [2], transformation and detec-tion. Bulk optics quantum gates are replaced by on-chip quan-tum gates. So far basic integrated quantum photonic gateshave been successfully designed and fabricated [3, 4]. Nev-ertheless, as quantum devices increase in size, the fabricationerrors become a problem and have to be kept under control.A unitary transformation U is usually decomposed in termsof simpler gates. These elementary gates are acting either on asingle optical mode (phase-shifts P ϕ ) or on two optical modes(beamsplitters) [5, 6]. To date, almost all optical experimentsuse this decomposition in terms of beamsplitters and variablephase-shifts [7].This decomposition is convenient since beamsplitters andphase-shifts can be straightforwardly implemented both inbulk optics and in integrated photonics. For example, chip-integrated photonics implement beamsplitters as multi-modeinterference devices and phase-shifts as heating metallic pads[8].However, this decomposition is not robust against perturba-tions, i.e., fabrication errors in beamsplitters and phase-shifts.To address this problem, a different decomposition of a n - mode unitary transformation has been proposed recently [9–11]. The new decomposition uses alternating layers of phase-shifts P k (the variable elements) and mixing transformations V k (the fixed elements) U = P V · · · P D V D P D +1 (1)The mixing transformations V k are acting globally an all n modes, in contrast to a local beamsplitter which and acts onlyon two neighbouring modes. The new decomposition (1) ismore robust to implementations errors, as the authors showedin Ref. [9].In this context one problem emerges, namely how to de-sign a photonic circuit which implements a given multi-modeunitary, like the mixing gate V k in (1). Here we propose an al-gorithm which addresses this problem. Given a n -port unitarytransformation U , our algorithm designs an integrated pho-tonic circuit which implements the transformation U .The structure of the article is the following. In Section IIwe describe the simulation algorithm, we discuss the mainansatz and the optimisation strategy. In section III we presentsimulations for different quantum gates: Hadamard H , 4-dimensional Fourier transform F and 2D random unitaries,together with error analysis. Finally, we conclude in SectionIV. II. FROM QUANTUM GATES TO PHOTONIC DEVICES
The problem we address here is the following. Given a uni-tary transformation U ∈ U( n ) acting on n spatial modes | out (cid:105) = U | in (cid:105) (2)our goal is to design a n -mode photonic device implementing U , see Fig. 1. In our case the n spatial modes are waveg-uides attached to the input (output) of the device. In termsof quantum information, the device implements a transforma-tion U over a n -dimensional qudit space. We use the path(spatial mode) representation for qudits, i.e., the basis state | i (cid:105) is represented by a (single-photon) wave-function in the i -thwaveguide of the device, ≤ i ≤ n − .In this section we discuss several necessary ingredients: (i)representation of the initial state; (ii) optimisation algorithm;(iii) fidelity measure. a r X i v : . [ qu a n t - ph ] J un U | in i | out i FIG. 1: A photonic device implementing the unitary transformation U on n optical modes (black input/output lines): | out (cid:105) = U | in (cid:105) . Initial state.
Since we use FDTD, which is a classical algo-rithm, to solve a quantum problem, we need to know how tomake the transition between classical and quantum descrip-tions. We consider the following ansatz for the electric field E of the photon wave-function travelling in the waveguides: E ( t ) = A σ √ π e − ( t − µσ ) sin( ωt + φ ) (3)i.e., a Gaussian-modulated sine-function and we consider thisas the classical description of a quantum wave-function ofphase φ . This is the transition from quantum to classical.Since the final classical states are mixed, we use an indi-rect approach to make the transition back to the quantum de-scription. We calculate the overlap between the target and thesimulated fields and we interpret this figure-of-merit as thequantum fidelity, as explained in the ’Fidelity’ subsection be-low. Optimisation.
The algorithm has several parts. First, we needto generate a photonic structure representing the device. Ourdevice consists of blocks (called pixels) which are either solid(silicon) or empty (air), see Fig. 2. Second, given a devicestructure, we propagate the input state through the device tofind out the output state. Finally, we need to optimise thestructure such that the device approximates as closely as pos-sible the transformation U .Our algorithm is based on iterative designs of classical op-tical devices [12] and direct binary search (DBS) [13, 14]. Westart with a solid block of silicon connected by n input and n output waveguides. We divide the active area of the deviceinto smaller blocks ( pixels ) which can be either on (filled withsilicon) or off (filled with air). We alternate between optimis-ing the structure and fine-graining until we reach the lowestpixel size which is still technologically feasible (Fig.2). Theoptimisation algorithm is presented in Annex A.Fine-graining is an essential step of our algorithm. Ini-tially we had developed only the optimisation part withoutfine-graining, by starting with the maximum number of pix-els (e.g., 64). However, we soon discovered that the algorithmwas prone to getting stuck in a local minimum, far from anoptimal value.We use a 2D finite-difference time-domain (FDTD) algo-rithm [15] and propagate the input state through the devicein order to obtain the output state. We record the values ofthe FDTD discrete electric and magnetic fields over the entiresimulation in a matrix (the ’test’ matrix). Fidelity.
Next we need to evaluate how close is the resultingquantum state compared to the desired (target) output state.We perform a separate simulation only with waveguides con-necting the input and the output; this is equivalent to a de-
FIG. 2: The structure of the simulation algorithm. The algorithmapplies iteratively to the silicon chip the following two steps: (1)optimisation, and (2) fine graining; black (grey) pixels are silicon(air). vice implementing the identity matrix I . We generate the tar-get output fields and record the discrete electric and magneticfields in a matrix (the ’target’ output). To compare the targetand test output we use the electromagnetic description of thecomplex wave function, see Ref. [16]: | Ψ (cid:105) = 1 √ (cid:34) −→ Ei −→ B (cid:35) (4)We define the fidelity (overlap) between the target | Ψ (cid:105) andtest output | Ψ (cid:105) as: F = 1 N |(cid:104) Ψ | Ψ (cid:105)| (5)and ≤ F ≤ with the norm N = (cid:112) (cid:104) Ψ | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) ; thefidelity is clearly symmetric in Ψ , Ψ . From eq. (4) we have: (cid:104) Ψ | Ψ (cid:105) = 12 ( −→ E · −→ E + −→ B · −→ B ) (6)We use transverse electric (TE) FDTD simulation, meaningone electric component on the z direction and two magneticcomponents on x and y . Taking in account that the fields arediscrete, the fidelity F becomes: F = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) grid pts E z (cid:12) E z + B x (cid:12) B x + B z (cid:12) B z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7)where (cid:12) is the Hadamard product and the sum is performedover all FDTD grid points. We interpret this as the quantumfidelity between the two states. III. RESULTS
In this section we present simulations for a single-qubit ap-plication, the Hadamard gate H , and a single-qudit applica-tion, the 4-dimensional Fourier gate F . For simulations weuse λ = 650 nm and (cid:15) r = 11 . (silicon).Importantly, FDTD simulations are scale invariant, due tothe scale invariance of Maxwell equations. This implies thatwe can arbitrarily choose the scale a of the system. By takingthe speed of light c = 1 , then a (or a/c ) defines the unit oftime. Consequently, the frequency is expressed in units of /a . We have chosen the scale a = 1 µ m corresponding to thesmallest possible pixel being 125 ×
125 nm, with λ = 650 nm.Clearly, every configuration (device) can be easily scaled fordifferent λ . FIG. 3: Propagation of a wave-function through the Hadamard gate.Each figure is a superposition of three snapshots: the initial state, thestate during propagation inside the gate and the output state. Left:simulation of the | (cid:105) state. Right: simulation of the | (cid:105) state. Noticethe π -phase difference in the output of the | (cid:105) state.FIG. 4: Fidelity F as a function of iteration steps for Hadamard H gate; F max = 0 . . A. Qubit
The Hadamard gate is defined as: H = 1 √ (cid:20) − (cid:21) (8)The two spatial modes correspond to the qubit basis states: | (cid:105) (top waveguide) and | (cid:105) (bottom waveguide).For the Hadamard gate, the electric field E in both outputwaveguides should have the same amplitude, but with differ-ent phases depending on the initial state: ϕ = 0 for the inputstate | (cid:105) and ϕ = π for the input state | (cid:105) . We clearly see thephase difference in the output electric field for the input state | (cid:105) , Fig.3.The fidelity increases with the number of iteration steps.Not surprisingly, the largest increase in fidelity occurs imme-diately after fine-graining, see Fig. 4. FIG. 5: Simulation of the 4-dimensional Fourier transform F . Asbefore, each figure is a superposition of three snapshots: the initialstate, the state during propagation inside the gate and the output state.The four panels correspond to the four basis states | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) .FIG. 6: Fidelity F as a function of iteration steps for the Fourier gate F ; F max = 0 . . B. Qudit
For our qudit application we chose a 4-dimensional Fouriergate F . The d -dimensional Fourier transform F d is [17]: F d | k (cid:105) = 1 √ d d − (cid:88) j =0 e πid kj | j (cid:105) (9)The Fourier F d generalises the Hadamard gate H for d > .As before, each waveguide corresponds to a basis state (topto bottom): | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) . Our design for the Fouriergate (Fig. 5) is compact, compared to previous implementa-tions which involved a large number of optical elements in acomplex design [18]. FIG. 7: Fidelity as a function of input wavelength. The vertical linecorresponds to 650 nm, the wavelength used during optimisation.
C. Random unitaries
To show that our algorithm works well for arbitrary gates,we generate structures for random unitaries and compute theirfidelity. We simulated 2 × ×
32 pixels, due to our limited computational power. Theunitary matrices are drawn from a uniform
U(2) distribution[19]: U ( α, φ, ψ, χ ) := e iα (cid:18) e iψ cos φ e iχ sin φ − e − iχ sin φ e − iψ cos φ (cid:19) (10)with the sampling φ ∈ [0 , π ] and α, ψ, χ ∈ [0 , π ] .We obtain an average fidelity F = 0 . ± . . This issimilar to the fidelity for the Hadamard gate, showing that ouralgorithm generates consistent results for arbitrary gates. D. Error analysis
Our goal is to design quantum gates which will be experi-mentally implemented. Thus it is important to know how fi-delity varies in practice with different sources of errors.First, we are interested in analysing the effects of variablephoton wavelength. Not surprisingly, fidelity is robust forwavelengths λ > λ larger than the optimised value λ , butdecreases rapidly for shorter ones, see Fig. 7.The second source of errors is manufacturing imprecision.To study this, we randomly shift each pixel relative to its orig-inal position. Significantly, fidelity is almost constant for dis-placement errors below 5 nm, then decreases almost linearlyfor larger values, Fig. 8. Thus if the fabrication errors are be-low 5 nm, the device will have a fidelity close to the simulatedone. FIG. 8: Fidelity as a function of pixel displacement.
IV. DISCUSSION
Integrated photonics is one of the most promising platformsfor future quantum technologies. In order to fully take ad-vantage of this platform, we need flexible tools to design andsimulate chip-integrated quantum gates.Here we presented an algorithm for designing integratedsilicon devices performing arbitrary quantum gates U on n spatial modes. Starting from a uniform block of silicon, the al-gorithm alternates optimisation and fine-graining steps in or-der to reach a photonic structure implementing U . We haveachieved fidelities up to 0.887 and we expect to surpass 0.9 byquadrupling the number of final pixels to 128 × photonic subroutines ,i.e., sets of quantum gates which are repeatedly used dur-ing the execution of a program. An example is the Fouriertransform F n on n modes. Usually F n is decomposed in n (log n +1) beamsplitters and n (log n − phase-shiftsand has optical depth d = n log n + 1 [20]. Thus it is moreefficient to have a specialised photonic circuit which performs F n in one step. This corresponds to optical depth 1, com-pared to the optical depth d for the standard decomposition interms of beamsplitters and phase-shifts.Another future application are dedicated quantum devices,similar to classical embedded systems. Examples are quan-tum communication, quantum sensing and quantum imagingdevices, where full programmability is not required. In thisscenario embedded quantum systems need to execute a partic-ular task fast and reliable without being fully programmable.Thus, a custom-designed photonic device which implementsin a single step a given unitary will be small, robust and fastcompared to a fully programmable processor. Acknowledgments
The authors acknowledge support from a grant of theRomanian Ministry of Research and Innovation, PCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0338/79PCCDI/2018, within PNCDI III. R.I. acknowledgessupport from PN 19060101/2019-2022.
Appendix A: Optimisation algorithm
A detailed flowchart of the optimisation algorithm is shownin Fig. 9. The optimisation algorithm starts by making a randomly or-dered list of pixels. Then it goes through each pixel and flipsits state. If the new fidelity F is higher, it keeps the pixelflipped. After testing every pixel, it compares the improve-ment of F across all steps. If this improvement is non-zero, itruns the DBS algorithm again. After the configuration cannotbe improved anymore, each pixel is subdivided into 4 squaresof the same type, which will be the new pixels, such that ifthe initial pixel was on (off) the new pixels are on (off). Wekeep optimising and fine graining until we reach a certain sizethreshold, given by fabrication constraints. [1] F. Flamini, N. Spagnolo, F. Sciarrino, Photonic quantum in-formation processing: a review , Rep.Prog.Phys. , 016001(2019).[2] J.W. Silverstone, D. Bonneau, M.J. Strain, M. Sorel,J.L. O’Brien, and M.G. Thompson, Silicon quantum photoniccircuits for on-chip qubit generation, manipulation and logicoperations, in Frontiers in Optics 2013, I. Kang, D. Reitze, N.Alic, and D. Hagan, eds., OSA Technical Digest (online) (Op-tical Society of America, 2013), paper FW4C.5.[3] E. Heemskerk and B.I. Akca,
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