Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Topologically-driven impossibility of superposing unknown states.

Zuzana Gavorová

Published Nov 3, 2021• 0 citations
OpenAlex

Abstract

Our understanding of quantum mechanics broadens by discovering tasks that are easy classically but their quantum versions are impossible, such as cloning. Another such task is the superposition task: Given two input states $\left|u\right>,\left|v\right>\in\mathcal{H}$ create a superposition $\alpha e^{i\phi(\left|u\right>,\left|v\right>)} \left|u\right>+\beta\left|v\right>$ for some real function $\phi$. This is the quantum version of the classical adder. Oszmaniec, Grudka, Horodecki and W\'ojcik [1] showed that a quantum circuit acting on a single pair of input states cannot output an exact superposition of them. Here we prove that a quantum circuit cannot approximate a superposition for every pair of states in $\mathcal{H}$ - even if it accesses any number of copies of the input states. Our proof follows the [2]: Any quantum circuit defines a continuous function on the input states. Using new topological arguments we prove that no continuous function can have the desired output. Compared to methods that view quantum circuits as linear [3-5] or polynomial [6] functions, the topological method excludes any number of input-state copies. We give some intuition of why the method is applicable to the superposition task. We consider the following protocol: given a large number of copies of the two input states, use state tomography to obtain a classical description of each, then prepare their superposition from scratch. Comparing our impossibility to this tomography protocol, we find a separation between two types of measurement.

Related papers

Topologically driven no-superposing theorem with a tight error bound

Zuzana Gavorová

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.

arXiv:2111.02391•Quantum•Nov 20, 2025•0 citations

Impossibility of creating a superposition of unknown quantum states

Somshubhro Bandyopadhyay

The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this Rapid Communication we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to the violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.

arXiv:2011.02275•Physical Review A•Nov 23, 2020•6 citations

Topological obstructions to quantum computation with unitary oracles

Zuzana Gavorová, Matan Seidel, Yonathan Touati

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 speedup. 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† 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 UdimU, UT , and U† algorithms. Our results strengthen an advantage of linear optics, challenge the experiments on relaxed causality, and motivate new algorithms with many-outcome measurements. Published by the American Physical Society 2024

arXiv:2011.10031•Physical Review A•Mar 28, 2024•4 citations

Why is it not possible to create a superposition of unknown quantum states

Somshubhro Bandyopadhyay

The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this paper we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.

•Nov 4, 2020•0 citations

Experimentally superposing two pure states with partial prior knowledge

Keren Li, Guofei Long, Hemant Katiyar +4 more

Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it was proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec et al., Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some given referential state. In this work, we experimentally realize the probabilistic protocol of superposing two pure states in a three-qubit nuclear magnetic resonance system. We demonstrate the feasibility of the protocol by preparing a families of input states, and the average fidelity between the prepared state and expected superposition state is over 99%. Moreover, we experimentally illustrate the limitation of the protocol that it is likely to fail or yields very low fidelity, if the nonzero overlaps are approaching zero. Our experimental implementation can be extended to more complex situations and other quantum systems.

arXiv:1608.04349•Physical Review A•Feb 23, 2017•19 citations

Optimal copy complexity of quantum state cloning

Sangwoo Jeon, Vaughn Sohn, Changhun Oh

Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for $d$-dimensional states of rank at most $r$: producing $M$ additional copies with worst-case fidelity at least $1-\varepsilon$ requires and is achievable with $N=Θ(Mrd/\varepsilon)$ input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For $M=1$, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.

quant-pharXiv:2608.24484•Aug 26, 2026•0 citations
Our understanding of quantum mechanics broadens by discovering tasks that are easy classically but their quantum versions are impossible, such as cloning. Another such task is the superposition task: Given two input states ∣u>,∣v>∈H\left|u\right>,\left|v\right>\in\mathcal{H}∣u⟩,∣v⟩∈H create a superposition αeiϕ(∣u>,∣v>)∣u>+β∣v>\alpha e^{i\phi(\left|u\right>,\left|v\right>)} \left|u\right>+\beta\left|v\right>αeiϕ(∣u⟩,∣v⟩)∣u⟩+β∣v⟩ for some real function ϕ\phiϕ. This is the quantum version of the classical adder. Oszmaniec, Grudka, Horodecki and W'ojcik [1] showed that a quantum circuit acting on a single pair of input states cannot output an exact superposition of them. Here we prove that a quantum circuit cannot approximate a superposition for every pair of states in H\mathcal{H}H - even if it accesses any number of copies of the input states. Our proof follows the [2]: Any quantum circuit defines a continuous function on the input states. Using new topological arguments we prove that no continuous function can have the desired output. Compared to methods that view quantum circuits as linear [3-5] or polynomial [6] functions, the topological method excludes any number of input-state copies. We give some intuition of why the method is applicable to the superposition task. We consider the following protocol: given a large number of copies of the two input states, use state tomography to obtain a classical description of each, then prepare their superposition from scratch. Comparing our impossibility to this tomography protocol, we find a separation between two types of measurement.
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.
The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this Rapid Communication we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to the violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.
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 speedup. 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† 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 UdimU, UT , and U† algorithms. Our results strengthen an advantage of linear optics, challenge the experiments on relaxed causality, and motivate new algorithms with many-outcome measurements. Published by the American Physical Society 2024
The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this paper we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.
Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it was proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec et al., Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some given referential state. In this work, we experimentally realize the probabilistic protocol of superposing two pure states in a three-qubit nuclear magnetic resonance system. We demonstrate the feasibility of the protocol by preparing a families of input states, and the average fidelity between the prepared state and expected superposition state is over 99%. Moreover, we experimentally illustrate the limitation of the protocol that it is likely to fail or yields very low fidelity, if the nonzero overlaps are approaching zero. Our experimental implementation can be extended to more complex situations and other quantum systems.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1M=1M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.