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Quantum Computer-Aided Design of Quantum Optics Hardware Quantum computer-aided design of quantum optics hardware Jakob S. Kottmann,1, 2, ∗ Mario Krenn,1, 2, 3, y Thi Ha Kyaw,1, 2 Sumner Alperin-Lea,1, 2 and Al´anAspuru-Guzik1, 2, 3, 4, z 1Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Canada. 2Department of Computer Science, University of Toronto, Canada. 3Vector Institute for Artificial Intelligence, Toronto, Canada. 4Canadian Institute for Advanced Research (CIFAR) Lebovic Fellow, Toronto, Canada (Dated: May 4, 2021) The parameters of a quantum system grow exponentially with the number of involved quantum particles. Hence, the associated memory requirement to store or manipulate the underlying wave- function goes well beyond the limit of the best classical computers for quantum systems composed of a few dozen particles, leading to serious challenges in their numerical simulation. This implies that the verification and design of new quantum devices and experiments are fundamentally limited to small system size. It is not clear how the full potential of large quantum systems can be exploited. Here, we present the concept of quantum computer designed quantum hardware and apply it to the field of quantum optics. Specifically, we map complex experimental hardware for high-dimensional, many-body entangled photons into a gate-based quantum circuit. We show explicitly how digital quantum simulation of Boson sampling experiments can be realized. We then illustrate how to design quantum-optical setups for complex entangled photonic systems, such as high-dimensional Greenberger-Horne-Zeilinger states and their derivatives. Since photonic hardware is already on the edge of quantum supremacy and the development of gate-based quantum computers is rapidly advancing, our approach promises to be a useful tool for the future of quantum device design. I. INTRODUCTION verify their correct execution when such calculations step beyond the point of classical calculation? Photonic systems are highly flexible and controllable for small to medium-sized quantum systems, and offer Here we illustrate a solution to solve the verification resilience against decoherence [1,2]. These properties and the design processes of quantum optical setups. make them a first choice in many proof-of-concepts We demonstrate how quantum optical systems can be in quantum information science. Examples include recast in the language of digital quantum computers observations of fundamental quantum properties, such and use the state-of-the-art simulators of quantum as indefinite causal orders [3], early demonstrations computers to design experiments for complex multi- of Wigner's friend paradox [4,5], high-dimensional photon entangled quantum systems. Furthermore, we quantum communication systems such as quantum key showcase the quantum simulation for one of the first distribution [6,7], entanglement swapping [8], quantum demonstrations of Boson sampling [32], illustrating that teleportation [9] and experimental quantum machine digital quantum computers can function as witnesses learning [10, 11] and new propositions for quantum for photonic quantum supremacy experiments that are technologies [12{17]. expected in the near future. While quantum experiments historically have been de- Due to the rapid progress in the development of gate signed by experienced human experts, their non-intuitive based quantum computers in the recent years [33, 34], nature has led to the emergence of computational we estimate that the design of photonic hardware with methods for designing quantum experiments [18{25]. quantum computers will become a realistic scenario in However, as the dimension of state space grows expo- the near future. In the meantime, the optimization nentially with the number of photons, this approach strategies presented here could also serve as valuable is limited to small systems. Consequently, while the benchmarks complementary to quantum chemistry [35]. abilities of photonic hardware constantly improve [26{ While the latter mainly focuses on determining an 30], there is no efficient computational method that can energy for an unknown ground state, the optimization of arXiv:2006.03075v2 [quant-ph] 3 May 2021 take advantage of the vast resources provided by these an optical setup focuses on determining its parameters systems. Furthermore, photonic quantum supremacy for a desired target quantum state. experiments are close to the point where they cannot be calculated with classical hardware [31]. How can one We propose the design and simulation of general quan- tum hardware as a new application for quantum com- puters. In this manuscript, we focus on photonic quan- ∗ [email protected] tum hardware by translating optical elements and mea- y [email protected] surement techniques into gate based quantum computers z [email protected] language. In a separate paper, some of us target the de- 2 sign of efficient superconducting qubit architectures by shifter acting on path mode a can then be written as translating the corresponding Hamiltonians into a digi- y tal quantum circuit [36]. PS (φ) = eia a (1) with the associated relative phase eiφ. With the mapping II. QUANTUM SIMULATION OF OPTICAL ELEMENTS of Fig.5 an n-qubit implementation of the phase shifter can be realized by a collection of single qubit operations In the following, we will explain the mapping from quan- n−qubits n−1 n−2 tum optics onto quantum circuits. A quantum optical P (φ) −−−−−−! S 2 φ ⊗ S 2 φ ⊗ :::S (φ) ;(2) setup consists of multiple optical path modes (paths) −i φ (σ −1) iφ which can be occupied by multiple photons with addi- where the S(φ) = e 2 z gate adds the phase e tional internal degrees of freedom (modes) like for ex- to the state j1i and leaves j0i invariant (see the ap- ample orbital angular momentum of light [37{39]. The pendix for more details). In the optimization of a quan- photonic occupation number of each internal degree of tum optical setup, we aim to optimize the fidelity of freedom is represented by a set of qubits. We will use the state, created by the setup, with a specific target binary encoding and we refer to the Ref. [40] for detailed state. In the next section (Eqs. (3) and (5)) we will analyses of other encodings (see also Refs. [41, 42] for construct this fidelity as a function of expectation values unary encodings). In this representation, the number of E = hHiU(φ) depending on the phase shifter parameter qubits needed to represent an optical setup is given by φ through the unitary U(φ) that encodes the quantum optical setup, and a Hamiltonian H that encodes the N = N × N × dlog (N )e; qubit modes paths 2 γ measurements. In order to compute the gradient of the fidelity with respect to the phase shifter parameter φ, we where Nγ is the maximum number of photons in one mode and we have used the integer ceiling function. With need the gradients of the expectation value @φE. Fol- this encoding, a basis state of the photonic setup can lowing Schuld et.al. [43] those gradients can be obtained N N with the parameter-shift rule, which allows the evalua- be represented as p mjnm;pi where p; m; n represent path, mode and number of photons, respectively. Take as tion of the analytical gradients via a finite-difference like procedure as @ E = E(φ − π ) + E(φ + π ). In order an example a setup with Npaths = 1 path, and Nmodes = 3 φ 4r 4r internal degrees of freedom, denoted as {−1; 0; 1g, where for this technique to be applicable, the generator of the parametrized quantum gate is required to have only two each can be occupied by up to Nγ = 3 photons. A state in distinct eigenvalues with distance 2r. Quantum gates this setup can then be represented by Nqubit = 6 qubits. Assuming that 2 of the photons occupy the mode −1 and that do not fulfill this condition can be decomposed into 1 photon occupies mode 1, the state can then be denoted more primitive gates, which allows to evaluate their gra- as dients by combining the parameter-shift rule with the product rule of calculus (see [46] for illustrations). In our qubits j2−1;a; 00;a; 1+1;ai −−−−!j10i−1;a ⊗ j00i0;a ⊗ j01i+1;a: explicit example the phase shifter is represented by a set 1 of S(φ) gates with the generator 2 (σz − 1). These gates These photonic states can be transformed by optical el- 1 fulfills the condition (r = 2 ) and no further decompo- ements which can be represented by digital quantum sition is necessary. Gradients of the other parametrized gates. In Fig.1 we show gate based representations of optical elements, such as the beam splitter or the herald- important optical elements for high-dimensional quan- ing process are obtained in an analogue fashion. The tum optics and provide further details in the appendix. parametrized part of the beam splitter is represented by single qubit rotations which fulfill the requirements for the parameter-shift rule and also do not require further A. Implementation decomposition. In our example in Fig.3 the heralding process contains a controlled rotation that needs to be One of the advantages of simulating the optical setups on further decomposed (see [46] for an explicit example). In a digital quantum computer is the direct access to gra- our implementation we use tequila [46], a high level dients of parametrized elements within a fully automat- python package that automatizes the illustrated gradi- ically differentiable framework. [43, 44] This fulfills all ent compilation and associated gate decompositions. We necessary conditions to replace the classical simulation refer to Refs. [46, 47] for more details on the implementa- module of topological optimizers such as theseus [45] tion of the automatically differentiable framework. Note as illustrated in Fig.2. We will illustrate the underlying that the evaluation of the gradient in this way would not framework here using the phase shifter of Fig.1 as an ex- be possible on an actual quantum optical setup, since the plicit example.
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