Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topologically driven no-superposing theorem with a tight error bound

Zuzana Gavorová

TL;DR

This work shows that a universal, deterministic quantum addition (i.e., a superposition) of two unknown quantum states cannot be implemented with arbitrary accuracy from copies, across standard quantum models, by a topology-based no-go argument. It quantifies a tight worst-case error bound $r$ and demonstrates tightness via a simple swap protocol, while distinguishing three computational models (trace-preserving, postselection, and random) with distinct capabilities for vector tomography. Vector tomography is impossible in the trace-preserving and postselection models but achievable in the random model, enabling a random protocol with exponential sample complexity; this reveals a qualitative separation between quantum computational models and highlights the special role randomness can play. The results deliver a deeper understanding of the limits of state preparation tasks, clarify the relationship between tomography and no-go theorems, and illustrate the applicability of topological methods to quantum information theory, while leaving open questions about optimal random-model protocols.

Abstract

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.

Topologically driven no-superposing theorem with a tight error bound

TL;DR

This work shows that a universal, deterministic quantum addition (i.e., a superposition) of two unknown quantum states cannot be implemented with arbitrary accuracy from copies, across standard quantum models, by a topology-based no-go argument. It quantifies a tight worst-case error bound and demonstrates tightness via a simple swap protocol, while distinguishing three computational models (trace-preserving, postselection, and random) with distinct capabilities for vector tomography. Vector tomography is impossible in the trace-preserving and postselection models but achievable in the random model, enabling a random protocol with exponential sample complexity; this reveals a qualitative separation between quantum computational models and highlights the special role randomness can play. The results deliver a deeper understanding of the limits of state preparation tasks, clarify the relationship between tomography and no-go theorems, and illustrate the applicability of topological methods to quantum information theory, while leaving open questions about optimal random-model protocols.

Abstract

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 of the superposition task, and it is tight. We present a trivial algorithm with a large approximation error and , and the matching impossibility of any smaller approximation error for any . 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.

Paper Structure

This paper contains 9 sections, 8 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Precise problem statement
  3. Overview of the results in different computational models
  4. Topology as a tool to prove no-go results
  5. The superposition impossibility
  6. State tomography for superposition
  7. Discussion
  8. Superposition impossibility via no-section
  9. Possibilites via randomness (proofs)

Key Result

Proposition 1

If a postselection algorithm takes as the input any number of copies of $\left|u\right\rangle\!\left\langle u\right|$ and $\left|v\right\rangle\!\left\langle v\right|$, then its output is a continuous function of $\left|u\right\rangle\!\left\langle u\right|, \left|v\right\rangle\!\left\langle v\righ

Figures (4)

  • Figure 1: (a) The probabilistic swap protocol swaps the two registers with probability $p=\sin^2\theta$ and does not swap them with probability $1-p=\cos^2\theta$. Then, the second register is traced out. (b) The controlled swap protocol has an additional control qubit to control the controlled swap and a control measurement to postselect the $\left|+\right\rangle=(\left|0\right\rangle+\left|1\right\rangle)/\sqrt{2}$ outcome.
  • Figure 2: The error as a function of the inputs for different protocols and $\alpha :\beta$ ratios. The controlled swap protocol (c-swap) outperforms the probabilistic swap protocol (p-swap) on almost all inputs, but the worst-case error of both is $r$.
  • Figure 3: Any closed loop on $\operatorname{S}^2$ is continuously contractible to a point. The same is true for $\operatorname{S}^3$ which is homeomorphic to $\hat{\mathbb{C}^2}$. We say that these spaces are simply connected.
  • Figure 4: The contradiction illustrated on $\operatorname{v}=\lim_{N\to\infty}f_N$ of \ref{['eq:v_example']}. This $\operatorname{v}=\operatorname{v}_0$ for a given matrix $\rho\in S\subset \mathcal{D}_\text{pure}(\mathcal{H})$ outputs that corresponding vector whose first entry is real positive, $\left\langle 0|\operatorname{v}_0(\rho)\right\rangle > 0$. The contradiction: functions $f_N$ should be both continuous and $(\epsilon_N+2\delta_N)$-close to $\operatorname{v}$ (i.e., confined to the marked regions).

Theorems & Definitions (21)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • proof
  • proof : Proof of \ref{['lem_2homog']}
  • Definition 2: Vector tomography
  • Theorem 2
  • proof
  • Definition 3: Vector tomography, revised
  • ...and 11 more