Scalable modern-time fault-tolerant quantum computation and quantum communication in a network employ a large number of physical qubits. For example, IBM is reported to have made a 127-qubit quantum computer. Unlike classical computation, quantum computation employs different types of logical qubits and qudits in terms of physical multiqubit and multiqudit systems respectively. Given this, of particular interest to us is to enquire on how quantum coherence in logical qubits is a manifestation of underlying quantum correlations in constituent physical multiqubit systems and vice-versa. In a recent work [Asthana, Sooryansh. New J Phys 24.5 (2022): 053026], we have shown that there is reciprocity in nonclassical correlations in physical multiqubit systems and coherence in a single logical qubit system. Subsequently, we have generalised the framework to higher dimensional quantum systems []. The crux of this study is that a single nonclassicality condition derived for quantum coherence in a logical system detects more than one type of nonclassicality in Hilbert spaces of nonidentical dimensions.
Scalable modern-time fault-tolerant quantum computation and quantum communication in a network employ a large number of physical qubits. For example, IBM is reported to have made a 127-qubit quantum computer. Unlike classical computation, quantum computation employs different types of logical qubits and qudits in terms of physical multiqubit and multiqudit systems respectively. Given this, of particular interest to us is to enquire on how quantum coherence in logical qubits is a manifestation of underlying quantum correlations in constituent physical multiqubit systems and vice-versa. In a recent work [Asthana, Sooryansh. New J Phys 24.5 (2022): 053026], we have shown that there is reciprocity in nonclassical correlations in physical multiqubit systems and coherence in a single logical qubit system. Subsequently, we have generalised the framework to higher dimensional quantum systems []. The crux of this study is that a single nonclassicality condition derived for quantum coherence in a logical system detects more than one type of nonclassicality in Hilbert spaces of nonidentical dimensions.
Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-3, distance-5 codewords, and general 2t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.
In a recent work [S. Asthana. New Journal of Physics 24.5 (2022): 053026], we have shown the interrelation of different nonclassical correlations in multiqubit systems with quantum coherence in a single logical qubit. In this work, we generalize it to higher-dimensional systems. For this, we take different choices of logical qudits and logical continuous-variable (cv) systems in terms of their constituent physical qudits and physical cv systems. Thereafter, we show reciprocity between conditions for coherence (in logical qudits and logical cv systems) and conditions for nonlocality and entanglement (in their underlying constituent qudits and cv systems). This shows that a single nonclassicality condition detects different types of nonclassicalities in different physical systems. Thereby, it reflects the interrelations of different nonclassical features of states belonging to Hilbert spaces of nonidentical dimensions.
Abstract As qubit-based platforms face near-term technical challenges in terms of scalability, qudits, d -level bases of quantum information, are being implemented in multiple platforms as an alternative for Quantum Information Processing (QIP). We compare the infidelity scalings of single qudit and multiqubit systems within identical Hilbert space dimensions and noisy environments in the Lindblad formalism. We find them to be gate-independent to first-order and present an analytically-derived critical curve (d2−1)/3log2(d) ( d 2 − 1 ) / 3 log 2 ( d ) that benchmarks the operational time efficiency of qudits and qubits relative to their decoherence times. This comparison reveals conditions under which qudits offer competitive gate efficiencies compared to leading qubit platforms. Our findings, supported by numerical simulations testing the applicability and limits of the linear response formalism, highlight the relevance of qudits in near-term QIP. This provides a benchmark for evaluating qudit platforms, specifically those with lower dimensionality, in terms of their operational efficiency relative to the qubit state-of-the-art.
Qudits provide a larger local Hilbert space than qubits and have recently attracted interest as building blocks for efficient fault-tolerant quantum computation (FTQC). In this work, we extend many-hypercube (MHC) codes, high-rate concatenated quantum codes, from qubits to qudits of prime dimension q. We construct the level-2 and level-3 MHC codes with parameters [[62,42,22]]q and [[63,43,23]]q, respectively, and generalize the level-by-level minimum-distance (LLMD) decoder originally proposed for the qubit MHC codes. In a qudit X-error model, we evaluate the code-capacity performance of the qudit MHC codes for dimensions up to q=13. We find that the error threshold increases monotonically with the local dimension q, reaching 10.0% for q=13, nearly double the value of 5.1% for qubits. Furthermore, for sufficiently large local dimensions, we observe a pronounced waterfall regime in which the logical error rate decreases substantially faster than expected from conventional distance-based estimates. To understand the origin of the observed qudit advantage, we analyze the decoding performance in terms of the entropy of the underlying q-ary noise channel. This analysis reveals a competition between the increased syndrome information available at larger q and the simultaneous growth of the typical physical error weight at fixed entropy. These results demonstrate that higher-dimensional quantum systems can substantially improve the performance of high-rate concatenated quantum codes and motivate further development of qudit-based FTQC architectures.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. For example, an electron, a spin 1/2 nucleus or two mutually orthogonal polarization states (horizontal and vertical, for example) of a single photon can be a qubit. We may also employ a two-dimensional subspace, such as the ground state and the first excited state, of a multi-dimensional Hilbert space, such as atomic energy levels. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. In any case, the two states are identified as the basis vectors, |0〉 and |1〉, of the Hilbert space so that a general single qubit state takes the form |ψ〉 = α|0〉 + β|1〉, where |α|2 + |β|2 = 1. A multiqubit state is expanded in terms of the tensor products of these basis vectors. Each qubit must be separately addressable. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional (qutrit) or, more generally, d-dimensional (qudit). A system may be made of several different kinds of qubits. Qubits in an ion trap quantum computer, for instance, may be defined as: (1) hyperfine/Zeeman sublevels in the electronic ground state of ions (2) a ground state and an excited state of a weakly allowed optical transition and (3) normal mode of ion oscillation. A similar scenario is also proposed for Josephson junction qubits, in which two flux qubits are coupled through a quantized LC circuit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.