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The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

Scott Aaronson

TL;DR

The notes survey the complexity of preparing quantum states and implementing unitaries, linking foundational quantum information to quantum money, black holes, and AdS/CFT. A unifying thread is the study of circuit complexity, state complexity, and their implications for cryptography, information security, and fundamental physics. Key contributions include framing unitary synthesis and state generation as complexity problems, analyzing QMA/QCMA and quantum advice, and connecting complexity growth to holographic wormhole volume via the AdS/CFT correspondence. The work highlights deep, cross-disciplinary connections and outlines numerous open problems in quantum complexity, quantum money security, and the computational limits of decoding black-hole information.

Abstract

These are lecture notes from a weeklong course in quantum complexity theory taught at the Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit complexity---i.e., the minimum number of gates needed to prepare a given quantum state or apply a given unitary transformation---as a unifying theme tying together several topics of recent interest in the field. Those topics include the power of quantum proofs and advice states; how to construct quantum money schemes secure against counterfeiting; and the role of complexity in the black-hole information paradox and the AdS/CFT correspondence (through connections made by Harlow-Hayden, Susskind, and others). The course was taught to a mixed audience of theoretical computer scientists and quantum gravity / string theorists, and starts out with a crash course on quantum information and computation in general.

The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

TL;DR

The notes survey the complexity of preparing quantum states and implementing unitaries, linking foundational quantum information to quantum money, black holes, and AdS/CFT. A unifying thread is the study of circuit complexity, state complexity, and their implications for cryptography, information security, and fundamental physics. Key contributions include framing unitary synthesis and state generation as complexity problems, analyzing QMA/QCMA and quantum advice, and connecting complexity growth to holographic wormhole volume via the AdS/CFT correspondence. The work highlights deep, cross-disciplinary connections and outlines numerous open problems in quantum complexity, quantum money security, and the computational limits of decoding black-hole information.

Abstract

These are lecture notes from a weeklong course in quantum complexity theory taught at the Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit complexity---i.e., the minimum number of gates needed to prepare a given quantum state or apply a given unitary transformation---as a unifying theme tying together several topics of recent interest in the field. Those topics include the power of quantum proofs and advice states; how to construct quantum money schemes secure against counterfeiting; and the role of complexity in the black-hole information paradox and the AdS/CFT correspondence (through connections made by Harlow-Hayden, Susskind, and others). The course was taught to a mixed audience of theoretical computer scientists and quantum gravity / string theorists, and starts out with a crash course on quantum information and computation in general.

Paper Structure

This paper contains 69 sections, 58 theorems, 165 equations, 6 figures.

Table of Contents

  1. Quantum Information Basics
  2. Introduction
  3. Basics of Quantum Mechanics
  4. Quantum States, Unitary Transformations, and Measurements
  5. Multipartite Systems, Entanglement, and Density Matrices
  6. More General Operations and Measurements
  7. Measuring Distances Between Quantum States
  8. "Almost As Good As New" Lemma and Quantum Union Bound
  9. Quantum Circuits, Gates, and Complexity
  10. The No-Cloning Theorem
  11. Monogamy of Entanglement
  12. Quantum Circuits and Universal Gate Sets
  13. The Solovay-Kitaev Theorem
  14. Relations to Complexity Classes
  15. Classical Simulation of Quantum Computation
  16. ...and 54 more sections

Key Result

Theorem 1.2.1

Suppose Alice and Bob share a bipartite state. Then nothing that Alice chooses to do (i.e. any combination of measurements and unitary transformations) can change Bob's local state, i.e., the density matrix describing Bob's state.

Figures (6)

  • Figure 1: Some relevant complexity classes, partially ordered by known inclusions.
  • Figure 2: The intermediate states in the relevant 2-dimensional subspace for amplitude amplification. This provides a "proof by picture" for why amplitude amplification works.
  • Figure 3: Simplified Post's Lattice
  • Figure 4: The inclusion lattice of reversible gate classes
  • Figure 5: Classification of Reversible Gates with Quantum Ancillas
  • ...and 1 more figures

Theorems & Definitions (107)

  • Theorem 1.2.1: No Communication Theorem
  • Lemma 1.3.1: "Almost As Good As New Lemma" aar:adv, closely related to "Gentle Measurement Lemma" gentle
  • proof
  • Lemma 1.3.2: "Gentle Measurement" gentle
  • Lemma 1.3.3: "Quantum Union Bound" (e.g. aar:qmaqpoly)
  • proof
  • Theorem 2.1.1
  • proof
  • proof
  • Theorem 2.3.1: Solovay-Kitaev (see solovaykitaev)
  • ...and 97 more