arXiv subjects

Zuzana Gavorová

Publications and source records attributed to Zuzana Gavorová.

3 recordsLinked to original sources

arXiv2025

Topologically driven no-superposing theorem with a tight error bound

To better understand quantum computation we can search for its limits or no-gos, especially if analogous limits do not appear in classical computation. Classical computation easily implements and extensively employs the addition of two bit strings, so here we study 'quantum addition': the superposition of two quantum states. We prove the impossibility of superposing two unknown states, no matter how many samples of each state are available. The proof uses topology; a quantum algorithm of any sample complexity corresponds to a continuous function, but the function required by the superposition task cannot be continuous by topological arguments. Our result for the first time quantifies the approximation error and the sample complexity $N$ of the superposition task, and it is tight. We present a trivial algorithm with a large approximation error and $N=1$, and the matching impossibility of any smaller approximation error for any $N$. Consequently, our results limit state tomography as a useful subroutine for the superposition. State tomography is useful only in a model that tolerates randomness in the superposed state. The optimal protocol in this random model remains open.

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arXiv2024

Topological obstructions to quantum computation with unitary oracles

Algorithms with unitary oracles can be nested, which makes them extremely versatile. An example is the phase estimation algorithm used in many candidate algorithms for quantum speed-up. The search for new quantum algorithms benefits from understanding their limitations: Some tasks are impossible in quantum circuits, although their classical versions are easy, for example, cloning. An example with a unitary oracle $U$ is the if clause, the task to implement controlled $U$ (up to the phase on $U$). In classical computation the conditional statement is easy and essential. In quantum circuits the if clause was shown impossible from one query to $U$. Is it possible from polynomially many queries? Here we unify algorithms with a unitary oracle and develop a topological method to prove their limitations: No number of queries to $U$ and $U^\dagger$ lets quantum circuits implement the if clause, even if admitting approximations, postselection and relaxed causality. We also show limitations of process tomography, oracle neutralization, and $\sqrt[\dim U]{U}$, $U^T$, and $U^\dagger$ algorithms. Our results strengthen an advantage of linear optics, challenge the experiments on relaxed causality, and motivate new algorithms with many-outcome measurements.

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arXiv2022

Notes on distinguishability of postselected computations

The framework of postselection is becoming more and more important in various recent directions in Quantum Computation research. Postselection renders simple computational models able to perform general quantum computation. This was first observed for the linear optics model [E. Knill, R. Laflamme, G. J. Milburn, Nature 409, 46 (2001)], and has since provided us with many near-term candidates for the quantum advantage, commuting computations [M. J. Bremner, R. Jozsa, D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] being the first. To facilitate the discussion of errors in the presence of postselection, we define and characterize trace-induced distance and diamond distance of postselected computations. We show counterexamples to simple properties that one would expect of any distance measure; the properties of convexity (when considering only the pure-state inputs would suffice), contractivity, and subadditivity of errors. On the positive side, we prove that certain weaker versions of contractivity and subadditivity and a number of other properties are preserved in the postselected setting. We achieve this via a "conversion lemma" that translates any inequality from the standard to the postselected setting.

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