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arXiv:2412.09672 · Physical Review A

Quantum pushforward designs

Jakub Czartowski, Karol Życzkowski

Published Mar 27, 2025• 2 citations
arXivHTMLPDF

Abstract

Designs, structures connected to averaging with respect to a given measure using finite sets of points, have proven themselves to be invaluable tools across the field of quantum information, finding their uses in state and process tomography, key distribution, and others. In this work, we introduce a concept of pushforward designs that allows us to obtain new structures from already existing ones by mapping them between the spaces, with specific examples including simplex designs and mixed state designs from complex projective designs. Based on the general concept, we put forward a structure called channel $[t,k]$-design, allowing for averaging over space of quantum channels for systems in contact with an environment of dimension $k$. Based on this notion, we introduce the concept of effective environment dimensionality ${k}^{*}$, which we estimate for the IBM Kyoto quantum computer to be below 2.2 for times up to $350\phantom{\rule{4pt}{0ex}}\textmu{}\text{s}$.

DOI
10.1103/physreva.111.032433

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Designs, structures connected to averaging with respect to a given measure using finite sets of points, have proven themselves to be invaluable tools across the field of quantum information, finding their uses in state and process tomography, key distribution, and others. In this work, we introduce a concept of pushforward designs that allows us to obtain new structures from already existing ones by mapping them between the spaces, with specific examples including simplex designs and mixed state designs from complex projective designs. Based on the general concept, we put forward a structure called channel [t,k][t,k][t,k]-design, allowing for averaging over space of quantum channels for systems in contact with an environment of dimension kkk. Based on this notion, we introduce the concept of effective environment dimensionality k∗{k}^{*}k∗, which we estimate for the IBM Kyoto quantum computer to be below 2.2 for times up to 350\textmus350\phantom{\rule{4pt}{0ex}}\textmu{}\text{s}350\textmus.
If H1H_1H1​ and H2H_2H2​ are finite-dimensional Hilbert spaces, a channel from H1H_1H1​ to H2H_2H2​ is a completely positive, linear map I\mathcal{I}I that takes the set of states S(H1)\mathcal{S}(H_1)S(H1​) for H1H_1H1​ to the set of states S(H2)\mathcal{S}(H_2)S(H2​) for H2H_2H2​. Corresponding to I\mathcal{I}I there is a unique dual map I∗\mathcal{I}^*I∗ from the set of effects E(H2)\mathcal{E}(H_2)E(H2​) for H2H_2H2​ to the set of effects E(H1)\mathcal{E}(H_1)E(H1​) for H1H_1H1​. We call I∗(b)\mathcal{I}^*(b)I∗(b) the effect bbb conditioned by I\mathcal{I}I and the set Ic=I∗(E(H2))\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))Ic=I∗(E(H2​)) the conditioned set of I\mathcal{I}I. We point out that Ic\mathcal{I}^cIc is a convex subeffect algebra of the effect algebra E(H1)\mathcal{E}(H_1)E(H1​). We extend this definition to the conditioning I∗(B)\mathcal{I}^*(B)I∗(B) for an observable BBB on H2H_2H2​ and say that an observable AAA is in Ic\mathcal{I}^cIc if A=I∗(B)A=\mathcal{I}^*(B)A=I∗(B) for some observable BBB. We show that Ic\mathcal{I}^cIc is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.
This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).
We extend the framework of quantum pushforward designs to the approximate setting, in which moment operators agree only up to finite precision. We first formulate a general transfer theorem for maps that act linearly at the level of moments. This separates direct applications, controlled by standard Schatten-norm estimates, from refinements that use additional tensorial structure. Dephasing is shown to be contractive and therefore sends an approximate projective design to a simplex design without increasing its error. Ordinary partial-trace estimates give immediate bounds for mixed-state and channel designs. We then prove a sharp Schatten-norm bound for the partial trace restricted to the totally symmetric subspace. This replaces the full-environment norm coefficient by one governed by a symmetric-subspace dimension and yields asymptotically tighter estimates for both mixed-state and channel designs. Numerical simulations for induced mixed-state designs are consistent with the resulting hierarchy of bounds.
Quantum computers have emerged as an alternative computer paradigm that will allow to solve some complex problems of large numbers that are not tractable for classical computers, e.g. factorization. Quantum computers also have the ability to simulate quantum mechanics in a real and direct way. The development of quantum computers faces many unique obstacles. In recent years, various quantum computer prototypes have been created by com- panies such as Google and IBM, on which we can run some quantum algorithms. But these prototypes are far from the quantum computers we envision and can function like today?s classical computers. It is said that we are in the time of Noisy Intermediate-Scale Quantum (NISQ) computers. This means that current quantum computers are very noisy and their size is very small compared to classical comput- ers. The challenge with small, noisy quantum devices running quantum algorithms is that due to the hardware limitations they face which must be respected, quantum algorithm needs to be modified before their execution on quantum devices. To solve this problem in quantum computers, a mapping process is needed, which is part of the compilation step that modifies the quantum algorithms to take into account all the limitations of the targeted quantum device. After mapping process, the resulting quantum circuit usually has a higher number of gates and (execution time) latency, decreasing the algorithm?s reliability. Different mapping solutions have been already proposed. Most of them are meant for a specific quan- tum processor and different in methodology, approach and features. In addition, they are usually only compared in terms of added gates (gate overhead) or added circuit execution time (latency overhead). No thorough comparative analysis of the different mapping solutions performance and features has been performed so far. In this thesis, we apply structured Design Space Exploration to the mapping of quantum circuits. By doing so, we will be able to have a deeper understanding of this process, make a thorough comparison of the mapper approaches and even develop optimized solutions for given applications and quantum devices. Moreover, we apply Design Space Exploration to the generation of quantum algorithms used as benchmarks and start to explore profiling them in a structured approach.
Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate ttt-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders ttt, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel ttt-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping noise), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest in moment calculations.
Any physical process can be represented as a quantum channel mapping an initial state to a final state. Hence it can be characterized from the point of view of communication theory, i.e., in terms of its ability to transfer information. Quantum information provides a theoretical framework and the proper mathematical tools to accomplish this. In this context the notion of codes and communication capacities have been introduced by generalizing them from the classical Shannon theory of information transmission and error correction. The underlying assumption of this approach is to consider the channel not as acting on a single system, but on sequences of systems, which, when properly initialized allow one to overcome the noisy effects induced by the physical process under consideration. While most of the work produced so far has been focused on the case in which a given channel transformation acts identically and independently on the various elements of the sequence (memoryless configuration in jargon), correlated error models appear to be a more realistic way to approach the problem. A slightly different, yet conceptually related, notion of correlated errors applies to a single quantum system which evolves continuously in time under the influence of an external disturbance which acts on it in a non-Markovian fashion. This leads to the study of memory effects in quantum channels: a fertile ground where interesting novel phenomena emerge at the intersection of quantum information theory and other branches of physics. A survey is taken of the field of quantum channels theory while also embracing these specific and complex settings.