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arXiv:quant-ph/0304050 · Physical Review Letters

Maximizing the Hilbert Space for a Finite Number of Distinguishable Quantum States

Andrew D. Greentree, S. G. Schirmer, F. Green, Lloyd C. L. Hollenberg, A. R. Hamilton, Robert G. Clark

Published Mar 5, 2004• 93 citations
arXivHTMLPDF

Abstract

Given a particular quantum computing architecture, how might one optimize its resources to maximize its computing power? We consider quantum computers with a number of distinguishable quantum states, and entangled particles shared between those states. Hilbert-space dimensionality is linked to nonclassicality and, hence, quantum computing power. We find that qutrit-based quantum computers optimize the Hilbert-space dimensionality and so are expected to be more powerful than other qudit implementations. In going beyond qudits, we identify structures with much higher Hilbert-space dimensionalities.

DOI
10.1103/physrevlett.92.097901

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Given a particular quantum computing architecture, how might one optimize its resources to maximize its computing power? We consider quantum computers with a number of distinguishable quantum states, and entangled particles shared between those states. Hilbert-space dimensionality is linked to nonclassicality and, hence, quantum computing power. We find that qutrit-based quantum computers optimize the Hilbert-space dimensionality and so are expected to be more powerful than other qudit implementations. In going beyond qudits, we identify structures with much higher Hilbert-space dimensionalities.
Classical computing architectures encode information into discrete binary digits residing in isolated regions of phase space. Quantum computing replaces this paradigm by leveraging complex state vectors defined within continuous, multi-dimensional Hilbert spaces. This paper provides a unified mathematical framework examining finite-dimensional Hilbert spaces in quantum information processing. It traces the structural scaling of quantum state spaces from single-qubit systems up through composite tensor product structures and multi-qubit registers. By detailing computational basis representations, state transformations, probability amplitudes, and multipartite entanglement classes, such as Bell, GHZ, and W states, this paper demonstrates how exponential dimensional scaling enables quantum parallelism while presenting computational barriers to classical simulation.
The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other $N\ensuremath{-}m$ qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3N/3N/3 qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than N/3N/3N/3 falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.
A major difficulty in quantum computation is the ability to implement fault tolerant computations, protecting information against undesired interactions with the environment. Stabiliser codes were introduced as a means to protect information when storing or applying computations in Hilbert spaces where the local dimension is fixed, i.e. in Hilbert spaces of the form (CD)⊗n({\mathbb C}^D)^{\otimes n}(CD)⊗n. If DDD is a prime power then one can consider stabiliser codes over finite fields \cite{KKKS2006}, which allows a deeper mathematical structure to be used to develop stabiliser codes. However, there is no practical reason that the subsystems should have the same local dimension and in this article we introduce a stabiliser formalism for mixed dimensional Hilbert spaces, i.e. of the form CD1⊗⋯⊗CDn{\mathbb C}^{D_1} \otimes \cdots \otimes {\mathbb C}^{D_n}CD1​⊗⋯⊗CDn​. More generally, we define and prove a Singleton bound for quantum error-correcting codes of mixed dimensional Hilbert spaces. We redefine entanglement measures for these Hilbert spaces and follow \cite{HESG2018} and define absolutely maximally entangled states as states which maximise this entanglement measure. We provide examples of absolutely maximally entangled states in spaces of dimensions not previously known to have absolutely maximally entangled states.
The primary resource for quantum computation is the Hilbert-space dimension. Whereas Hilbert space itself is an abstract construction, the number of dimensions available to a system is a physical quantity that requires physical resources. Avoiding a demand for an exponential amount of these resources places a fundamental constraint on the systems that are suitable for scalable quantum computation. To be scalable, the number of degrees of freedom in the computer must grow nearly linearly with the number of qubits in an equivalent qubit-based quantum computer. These considerations rule out quantum computers based on a single particle, a single atom, or a single molecule consisting of a fixed number of atoms or on classical waves manipulated using the transformations of linear optics.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud; 03.65.Bz Quantum entanglement is an important resource to be implemented in quantum information processing and quantum communication protocols. [1] The interests in studying multi-dimensional entangled states come from both the foundations of quantum mechanics and the development of new protocols in quantum communication. For example, it was shown that maximally entangled states of two quantum systems in a high-dimensional Hilbert space, qudits, violate local realism stronger than qubits, and entangled qudits are less affected by noise than entangled qubits. [2,3] In quantum cryptography [4], the use of entangled qutrits [4,5] or qudits [6,7] instead of qubits is more secure against eavesdropping attacks. Furthermore, the protocols for quantum teleportation or for quantum cryptography, work best with maximally entangled states. All these facts motivate the development of techniques to generate entangled states among quantum systems in higher dimensional Hilbert space with a good quality of entanglement.
The Hilbert space of a physical qubit typically features more than two energy levels. Using states outside the qubit subspace can provide advantages in quantum computation. To benefit from these advantages, individual states of the d -dimensional qudit Hilbert space have to be discriminated during readout. We propose and analyze two measurement strategies that improve the distinguishability of transmon qudit states. Based on a model describing the readout of a transmon qudit coupled to a resonator, we identify the regime in hardware parameter space where each strategy is optimal. We discuss these strategies in the context of a practical implementation of the default measurement of a ququart on IBM Quantum hardware whose states are prepared by employing higher-order X gates that make use of two-photon transitions. Published by the American Physical Society 2024