We consider the problem of encoding classical information into unknown qudit states belonging to any basis, of a maximal set of mutually unbiased bases, by one party and then decoding by another party who has perfect knowledge of the basis. Working with qudits of prime dimensions, we point out a no-go theorem that forbids shift operations on arbitrary unknown states. We then provide the necessary conditions for reliable encoding/decoding.
We give a full explanation of the noiseless subsystem that protects a single-qubit against collective errors and the corresponding recursive scheme described by C.-K. Li et. al. [Phys. Rev. A 84, 044301 (2011)] from a representation theory point of view. Furthermore, we extend the construction to qudits under the influence of collective SU($d$) errors. We find that under this recursive scheme, the asymptotic encoding rate is $1/d$.
All known qudit-based prepare-and-measure quantum key distribution (PM-QKD) schemes are more error resilient than their qubit-based counterparts. Their high error resiliency comes partly from the careful encoding of multiple bits of signals used to generate the raw key in each transmitted qudit so that the same eavesdropping attempt causes a higher bit error rate (BER) in the raw key. Here I show that highly error-tolerant PM-QKD schemes can be constructed simply by encoding one bit of classical information in each transmitted qudit in the form $(|i\rangle\pm|j\rangle)/\sqrt{2}$, where $|i\ran…
Fault-Tolerant Resource Comparison of Qudit and Qubit Encodings for Diagonal Quadratic Operators
University of Liverpool（利物浦大学）；Fermi National Accelerator Laboratory（费米国家加速器实验室）；University of Trento（特伦托大学）；INFN-TIFPA Trento Institute of Fundamental Physics and Applications（意大利国家核物理研究所-特伦托基础物理与应用研究所）
Molecular Nanomagnets may enable the implementation of qudit-based quantum error-correction codes which exploit the many spin levels naturally embedded in a single molecule, a promising step towards scalable quantum processors. To fully realize the potential of this approach, a microscopic understanding of the errors corrupting the quantum information encoded in a molecular qudit is essential, together with the development of tailor-made quantum error correction strategies. We address these central points by first studying dephasing effects on the molecular spin qudit produced by the interacti…
The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of quantum systems is both of fundamental and practical interest. Understanding the invariance of a set of states under certain transformations is mutually associated with a better understanding of some fundamental aspects of quantum mechanics as well as the practical utility of invariant subsystems. For example, DFS/NSs are potentially useful for protecting quantum information in quantum cryptography and quantum computing as well as enabling universal computation. Here we discuss transformations which are compatible with a DF…
We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computation is adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of…
A new version of quantum hashing technique is developed wherein a quantum hash is constructed as a sequence of single-photon high-dimensional states (qudits). A proof-of-principle implementation of the high-dimensional quantum hashing protocol using orbital-angular momentum encoding of single photons is implemented. It is shown that the number of qudits decreases with increase of their dimension for an optimal ratio between collision probability and decoding probability of the hash. Thus, increasing dimension of information carriers makes quantum hashing with single photons more efficient.
The probabilistic nature of single-photon sources and photon-photon interactions encourages encoding as much quantum information as possible in every photon for the purpose of photonic quantum information processing. Here, by encoding high-dimensional units of information (qudits) in time and frequency degrees of freedom using on-chip sources, we report deterministic two-qudit gates in a single photon with fidelities exceeding 0.90 in the computational basis. Constructing a two-qudit modulo SUM gate, we generate and measure a single-photon state with non-separability between time and frequency…
We present a gauge invariant digitization of $(1+1)$d scalar quantum electrodynamics for an arbitrary spin truncation for qudit-based quantum computers. We provide a construction of the Trotter operator in terms of a universal qudit-gate set. The cost savings of using a qutrit based spin-1 encoding versus a qubit encoding are illustrated. We show that a simple initial state could be simulated on current qutrit based hardware using noisy simulations for two different native gate set.
Time-bin qudits have emerged as a promising encoding platform in many quantum photonic applications. However, the requirement for efficient single-shot measurement of time-bin qudits instead of reconstructive detection has restricted their widespread use in experiments. Here, we propose an efficient method to measure arbitrary superposition states of time-bin qudits and confirm it up to dimension 4. This method is based on encoding time bins at the picosecond time scale, also known as ultrafast time bins. By doing so, we enable the use of robust and phase-stable single spatial mode temporal in…
We propose a method to experimentally demonstrate contextuality with a family of tests for qudits. The experiment we propose uses a qudit encoded in the path of a single photon and its temporal degrees of freedom. We consider the impact of noise on the effectiveness of these tests, taking the approach of ontologically faithful non-contextuality. In this approach, imperfections in the experimental set up must be taken into account in any faithful ontological (classical) model, which limits how much the statistics can deviate within different contexts. In this way we bound the precision of the e…
We introduce a new quantum communication protocol for the transmission of quantum information under collective noise. Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented. The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, m. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described …
We prove that universal quantum computation can be realized---using only linear optics and $χ^{(2)}$ (three-wave mixing) interactions---in any $(n+1)$-dimensional qudit basis of the $n$-pump-photon subspace. First, we exhibit a strictly universal gate set for the qubit basis in the one-pump-photon subspace. Next, we demonstrate qutrit-basis universality by proving that $χ^{(2)}$ Hamiltonians and photon-number operators generate the full $\mathfrak{u}(3)$ Lie algebra in the two-pump-photon subspace, and showing how the qutrit controlled-$Z$ gate can be implemented with only linear optics and $χ…
Randomized encoding is a powerful cryptographic primitive with various applications such as secure multiparty computation, verifiable computation, parallel cryptography, and complexity lower-bounds. Intuitively, randomized encoding $\hat{f}$ of a function $f$ is another function such that $f(x)$ can be recovered from $\hat{f}(x)$, and nothing except for $f(x)$ is leaked from $\hat{f}(x)$. Its quantum version, quantum randomized encoding, has been introduced recently [Brakerski and Yuen, arXiv:2006.01085]. Intuitively, quantum randomized encoding $\hat{F}$ of a quantum operation $F$ is another …
High-dimensional quantum states, or qudits, represent a promising resource in the quantum information field. Here we present the experimental generation of four-dimensional quantum states, or ququarts, encoded in the polarization and orbital angular momentum of a single photon. Our novel technique, based on the q-plate device, allows to prepare and measure the ququart in all five mutually unbiased bases. We report the reconstruction of the four dimensional density matrix through the tomographic procedure for different ququart states.
By encoding a qudit in a harmonic oscillator and investigating the d --> infinity limit, we give an entirely new realization of continuous-variable quantum computation. The generalized Pauli group is generated by number and phase operators for harmonic oscillators.
We study the correlation clustering problem using the quantum approximate optimization algorithm (QAOA) and qudits, which constitute a natural platform for such non-binary problems. Specifically, we consider a neutral atom quantum computer and propose a full stack approach for correlation clustering, including Hamiltonian formulation of the algorithm, analysis of its performance, identification of a suitable level structure for ${}^{87}{\rm Sr}$ and specific gate design. We show the qudit implementation is superior to the qubit encoding as quantified by the gate count. For single layer QAOA, w…
A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-$s$, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by $d$-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of S…
By encoding a qudit in a harmonic oscillator and investigating the infinite limit, we give an entirely new realization of continuous-variable quantum computation. The generalized Pauli group is generated by number and phase operators for harmonic oscillators. We describe a physical realization in terms of modes in a microwave cavity, coupled via a standard Kerr nonlinearity.
