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Efficient Simulation of Random Quantum States and Operators

Christoph Dankert

TL;DR

This thesis develops practical methods for simulating and characterizing randomness in quantum systems. It establishes that Mutually-Unbiased Bases form a $2$-design, enabling Haar-average fidelity estimates with scalable circuits, and provides explicit constructions to generate MUB states efficiently. It also links unitary $2$-designs to the Clifford group, offering constructive proofs and practical implementations for sampling-based noise estimation. Collectively, the work unifies several notions of pseudo-randomness (2-designs, Haar measure, Clifford symmetry) with actionable protocols for fidelity estimation and noise characterization in quantum devices.

Abstract

We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.

Efficient Simulation of Random Quantum States and Operators

TL;DR

This thesis develops practical methods for simulating and characterizing randomness in quantum systems. It establishes that Mutually-Unbiased Bases form a -design, enabling Haar-average fidelity estimates with scalable circuits, and provides explicit constructions to generate MUB states efficiently. It also links unitary -designs to the Clifford group, offering constructive proofs and practical implementations for sampling-based noise estimation. Collectively, the work unifies several notions of pseudo-randomness (2-designs, Haar measure, Clifford symmetry) with actionable protocols for fidelity estimation and noise characterization in quantum devices.

Abstract

We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.

Paper Structure

This paper contains 97 sections, 56 theorems, 433 equations, 28 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preface
  3. Quantum Mechanics Framework
  4. Unitary Evolution
  5. Measurement
  6. Quantum Computing
  7. Turing Machine Model
  8. Circuit Model
  9. Quantum Algorithms
  10. Amplitude Amplification
  11. Noise in Quantum Computation
  12. Noise in the Classical World
  13. Noise in Quantum Computing
  14. Quantum Operations Formalism
  15. Single Qubit Noise
  16. ...and 82 more sections

Key Result

Theorem 1.3.1

Papadimitriou1994 A language $L \subset \{0,1\}^*$ can be computed by a uniformly polynomial circuit familiy iff $L \in \mathbf{P}$.

Figures (28)

  • Figure 1: Classical circuit computing $a \oplus b$
  • Figure 2: A quantum circuit that computes $a \oplus b$
  • Figure 3: The Subspace of Good and Bad States in Amplitude Amplification.
  • Figure 4: Amplitude Amplification as Two Reflections.
  • Figure 5: The Bit-Flip Error Model.
  • ...and 23 more figures

Theorems & Definitions (206)

  • Theorem 1.3.1
  • Definition 1.3.2
  • Lemma 1.3.3
  • proof
  • Theorem 1.3.4
  • Definition 2.2.1
  • Definition 2.3.2
  • Definition 2.3.5
  • Definition 2.3.7
  • Definition 2.3.8
  • ...and 196 more