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arXiv:2112.06549 · Physical Review Letters

Generating Haar-Uniform Randomness Using Stochastic Quantum Walks on a Photonic Chip

Hao Long Tang, Leonardo Banchi, Tian-Yu Wang, Xiao-Wen Shang, Xi Tan, Wen-Hao Zhou, Zhen Feng, Anurag Pal, Hang Li, Cheng-Qiu Hu, M. S. Kim, Xian-Min Jin

Published Feb 3, 2022• 30 citations
arXivHTMLPDF

Abstract

As random operations for quantum systems are intensively used in various quantum information tasks, a trustworthy measure of the randomness in quantum operations is highly demanded. The Haar measure of randomness is a useful tool with wide applications, such as boson sampling. Recently, a theoretical protocol was proposed to combine quantum control theory and driven stochastic quantum walks to generate Haar-uniform random operations. This opens up a promising route to converting classical randomness to quantum randomness. Here, we implement a two-dimensional stochastic quantum walk on the integrated photonic chip and demonstrate that the average of all distribution profiles converges to the even distribution when the evolution length increases, suggesting the 1-pad Haar-uniform randomness. We further show that our two-dimensional array outperforms the one-dimensional array of the same number of waveguide for the speed of convergence. Our Letter demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques.

DOI
10.1103/physrevlett.128.050503

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As random operations for quantum systems are intensively used in various quantum information tasks, a trustworthy measure of the randomness in quantum operations is highly demanded. The Haar measure of randomness is a useful tool with wide applications, such as boson sampling. Recently, a theoretical protocol was proposed to combine quantum control theory and driven stochastic quantum walks to generate Haar-uniform random operations. This opens up a promising route to converting classical randomness to quantum randomness. Here, we implement a two-dimensional stochastic quantum walk on the integrated photonic chip and demonstrate that the average of all distribution profiles converges to the even distribution when the evolution length increases, suggesting the 1-pad Haar-uniform randomness. We further show that our two-dimensional array outperforms the one-dimensional array of the same number of waveguide for the speed of convergence. Our Letter demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques.
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Quantum walks, in virtue of the coherent superposition and quantum interference, have exponential superiority over their classical counterpart in applications of quantum searching and quantum simulation. The quantum-enhanced power is highly related to the state space of quantum walks, which can be expanded by enlarging the photon number and/or the dimensions of the evolution network, but the former is considerably challenging due to probabilistic generation of single photons and multiplicative loss. We demonstrate a two-dimensional continuous-time quantum walk by using the external geometry of photonic waveguide arrays, rather than the inner degree of freedoms of photons. Using femtosecond laser direct writing, we construct a large-scale three-dimensional structure that forms a two-dimensional lattice with up to 49 × 49 nodes on a photonic chip. We demonstrate spatial two-dimensional quantum walks using heralded single photons and single photon-level imaging. We analyze the quantum transport properties via observing the ballistic evolution pattern and the variance profile, which agree well with simulation results. We further reveal the transient nature that is the unique feature for quantum walks of beyond one dimension. An architecture that allows a quantum walk to freely evolve in all directions and at a large scale, combining with defect and disorder control, may bring up powerful and versatile quantum walk machines for classically intractable problems.
Random unitary operations are a fundamental resource for quantum information processing, yet provably generating them with experimentally available controls remains challenging. Existing approaches often require calculating higher-order operator expectation values, relying on statistical assumptions about the experimental Hamiltonian's spectrum to make these calculations tractable. We show how to implement random unitaries by a sequence of evolutions generated by a fixed Hamiltonian conjugated by a global control operator. We give a geometric argument that establishes exponential convergence to the uniform random (Haar) measure on the accessible unitary group under uniform parameter sampling, beyond a finite sequence-length threshold that we bound. We show this holds when the connected Lie subgroup generated by the conjugated Hamiltonians is closed. Native dynamics and global control thus provide a constructive route to uniform randomness, with applications to many experimental platforms including Rydberg atoms, bosons and fermions in optical lattices, transmons, and trapped ions.
Abstract We study the sampling complexity of a probability distribution associated with an ensemble of identical noninteracting bosons undergoing a quantum random walk on a one-dimensional lattice. With uniform nearest-neighbor hopping we show that one can efficiently sample the distribution for times logarithmic in the size of the system, while for longer times there is no known efficient sampling algorithm. With time-dependent hopping and optimal control, we design the time evolution to approximate an arbitrary Haar-random unitary map analogous to that designed for photons in a linear optical network. This approach highlights a route to generating quantum complexity by optimal control only of a single-body unitary matrix. We study this in the context of two potential experimental realizations: a spinor optical lattice of ultracold atoms and a quantum gas microscope.
Abstract Quantum random number generation (QRNG) harnesses the intrinsic randomness of quantum mechanical phenomena. On-chip photonic circuitry provides a robust and versatile platform that can address and explore fundamental questions in quantum as well as classical physics. Likewise, integrated waveguide-based architectures hold the potential for intrinsically scalable, efficient and compact implementations of photonic QRNG. Here, we harness the quantum emission from the two-dimensional material hexagonal boron nitride an emerging atomically thin medium that can generate single photons on demand while operating at room temperature. By means of a customized splitter arrangement, we achieve true random number generation through the measurement of single photons exiting one of four designated output ports, and subsequently verify the randomness of the sequences in accordance with the National Institute of Standards and Technology benchmark suite. Our results clearly demonstrate the viability and efficiency of this approach to on-chip deterministic random number generators.
Quantum walks in an elaborately designed graph are a powerful tool for simulating physical and topological phenomena, constructing novel quantum algorithms, and realizing universal quantum computing. Integrated photonics technology has emerged as a versatile platform for implementing a variety of quantum information tasks and as a promising candidate for performing large-scale quantum walks. Both extending physical dimensions and involving more particles will increase the complexity of the evolving systems. Pioneering studies have demonstrated a single particle walking on two-dimensional lattices and multiple walkers interfering on a one-dimensional structure. However, multiple particles evolving in a genuine two-dimensional space in a scalable fashion has remained a vacancy for nearly 10 years. We present a genuine two-dimensional quantum walk with correlated photons on a triangular photonic lattice, which is mapped to a 37 × 37 high-dimensional state space. The genuine two-dimensional quantum walk breaks through the physical restrictions of single-particle evolution, allowing for the encoding of information in large spaces and construction of high-dimensional graphs, which are beneficial for quantum information processing. Between the chip and the two-dimensional fanout interface, site-by-site addressing enables simultaneous detection of over 600 nonclassical interferences and observation of quantum correlations that violate a classical limit by 57 standard deviations. Our implementation provides a paradigm for multi-photon quantum walks in a two-dimensional configuration on a large scale, paving the way for practical quantum simulation and computation beyond the classical regime.