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Structure, Optimality, and Symmetry in Shadow Unitary Inversion

Guocheng Zhen, Yu-Ao Chen, Mingrui Jing, Jingu Xie, Ranyiliu Chen, Xin Wang

TL;DR

This work introduces shadow unitary inversion, a framework that reverses unknown quantum dynamics only insofar as the inverse action reproduces the expectation value of a fixed observable, enabling substantially fewer queries than full inversion. It proves a dimension-dependent linear lower bound on the required number of queries, with the constant determined by the observable's spectrum, and provides a constructive three-query protocol for qubits along with a complete channel-characterization under the shadow constraint. For higher dimensions, it develops a symmetry-enhanced SDP framework based on Schur–Weyl duality that block-diagonalizes the optimization, reducing variable counts from $d^{4t+4}$ to at most $(t+1)!t! d^{t+1}$ and making scalable exploration feasible. Together, the results establish the first systematic framework for shadow unitary inversion, clarifying resource costs and symmetry structure across dimensions and offering practical tools for quantum control and information recovery tasks.

Abstract

Reversing unitary operations is a key task in quantum computing and quantum control. In this work, we introduce and develop the framework of shadow unitary inversion, a relaxed variant of unitary inversion in which the goal is to reproduce the action of the inverse unitary only at the level of the expectation value of a fixed observable. This task captures an operational setting in which only shadow information is required and allows query complexities significantly below those of full unitary inversion. We establish a dimension-dependent lower bound showing that any $t$-query scheme requires $t$ to scale at least linearly with the system dimension, with the constant determined by the spectral properties of the target observable. In the qubit case, we construct a deterministic three-query sequential protocol that achieves exact shadow inversion, and we provide a complete characterization of all admissible qubit channels satisfying the shadow constraint. Numerical evidence suggests that three queries are optimal. For higher-dimensional systems, we develop a semidefinite-programming formulation for optimizing shadow-inversion combs and introduce a representation-theoretic symmetry reduction that decomposes the problem into invariant blocks, substantially reducing the problem size. These results provide the first systematic study for shadow unitary inversion and establish its resource requirements and symmetry structure across dimensions.

Structure, Optimality, and Symmetry in Shadow Unitary Inversion

TL;DR

This work introduces shadow unitary inversion, a framework that reverses unknown quantum dynamics only insofar as the inverse action reproduces the expectation value of a fixed observable, enabling substantially fewer queries than full inversion. It proves a dimension-dependent linear lower bound on the required number of queries, with the constant determined by the observable's spectrum, and provides a constructive three-query protocol for qubits along with a complete channel-characterization under the shadow constraint. For higher dimensions, it develops a symmetry-enhanced SDP framework based on Schur–Weyl duality that block-diagonalizes the optimization, reducing variable counts from to at most and making scalable exploration feasible. Together, the results establish the first systematic framework for shadow unitary inversion, clarifying resource costs and symmetry structure across dimensions and offering practical tools for quantum control and information recovery tasks.

Abstract

Reversing unitary operations is a key task in quantum computing and quantum control. In this work, we introduce and develop the framework of shadow unitary inversion, a relaxed variant of unitary inversion in which the goal is to reproduce the action of the inverse unitary only at the level of the expectation value of a fixed observable. This task captures an operational setting in which only shadow information is required and allows query complexities significantly below those of full unitary inversion. We establish a dimension-dependent lower bound showing that any -query scheme requires to scale at least linearly with the system dimension, with the constant determined by the spectral properties of the target observable. In the qubit case, we construct a deterministic three-query sequential protocol that achieves exact shadow inversion, and we provide a complete characterization of all admissible qubit channels satisfying the shadow constraint. Numerical evidence suggests that three queries are optimal. For higher-dimensional systems, we develop a semidefinite-programming formulation for optimizing shadow-inversion combs and introduce a representation-theoretic symmetry reduction that decomposes the problem into invariant blocks, substantially reducing the problem size. These results provide the first systematic study for shadow unitary inversion and establish its resource requirements and symmetry structure across dimensions.

Paper Structure

This paper contains 18 sections, 27 theorems, 313 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. General formulation and fundamental limit
  3. Qubit shadow unitary inversion
  4. Symmetry-based SDP simplification for general dimension
  5. Concluding remarks
  6. Measure theory and representation theory preliminaries
  7. Quantum comb
  8. Qubit shadow unitary inversion
  9. A circuit implementation
  10. Necessary and sufficient condition
  11. A General Lower Bound on Query Complexity
  12. Comparison with the virtual comb approach
  13. General SDP framework for shadow inversion
  14. General SDP formulation and symmetry property
  15. The number of variables in the simplified SDP
  16. ...and 3 more sections

Key Result

Theorem 2

For any $d,t \in \mathbb{N+}$, if $\mathcal{N}$ is a $t$-query shadow inversion of $d$-dimensional unitaries under some fixed non-trivial $d$-dimensional Hermitian observable $O \not\propto I_{d}$, then we have the following lower-bound where $\lambda_{1} \geq \cdots \geq \lambda_{d}$ are the eigenvalues of $O$.

Figures (3)

  • Figure 1: The general framework of shadow inversion problem with respect to the unknown physical dynamics. With the access of $t$-query of the unknown unitary evolution $U$, our quantum device target to reproduce the shadow information of the initial state $\rho_0$ with respect to the observable $O$.
  • Figure S1: Three kinds of quantum combs involving the parallel, sequential and indefinite causal order. The alphabets $P,I_j,O_j$, and $F$ label the corresponding Hilbert spaces $\mathcal{H}_P, \mathcal{H}_{I_j},\mathcal{H}_{O_j}$, and $\mathcal{H}_F$, respectively.
  • Figure S2: The circuit configuration of decomposed quantum comb for inversing unknown single-qubit unitary regarding the observable $Z$. Three ancilla qubits are employed (initiated to $\ket{0}$); one acts as a catalytic ancilla that is restored to $\ket0$ (i.e., not consumed) by the end of the circuit.

Theorems & Definitions (62)

  • Definition 1: Shadow unitary inversion
  • Theorem 2
  • Theorem 3
  • Proposition 3
  • Theorem 4
  • Definition S1: Haar measure
  • Lemma S2: Tonelli’s theorem
  • Definition S3: Finite group representation
  • Definition S4: Irreducible representation
  • Definition S5: Partition of a natural number
  • ...and 52 more