Pseudo-randomness and Learning in Quantum Computation

TL;DR

The thesis addresses how to replace Haar randomness in quantum computing with efficient pseudo-random constructions. It develops a two-pronged approach: (i) establish that short random quantum circuits rapidly approximate a $2$-design, and (ii) engineer an efficient, scalable construction of unitary $k$-designs for general $k$ via quantum tensor product expanders (TPEs), leveraging a classical $2k$-TPE and a quantum Fourier transform. The work also develops learning and testing algorithms for the Clifford group, including an optimal procedure to identify an unknown Clifford operation and a tester to distinguish Clifford-like behavior from non-Clifford. The results unify several notions of approximate designs (diamond, trace, and monomial-based) and provide concrete bounds on mixing times, spectral gaps, and the number of random gates required. These contributions enable practical derandomisation and provide tools for analyzing concentration and large-deviation phenomena in Haar-like quantum processes, with immediate implications for quantum algorithms, cryptographic protocols, and quantum learning theory.

Abstract

This thesis discusses the young fields of quantum pseudo-randomness and quantum learning algorithms. We present techniques for derandomising algorithms to decrease randomness resource requirements and improve efficiency. One key object in doing this is a k-design, which is a distribution on the unitary group whose kth moments match those of the unitarily invariant Haar measure. We show that for a natural model of a random quantum circuit, the distribution of random circuits quickly converges to a 2-design. We then present an efficient unitary k-design construction for any k, provided the number of qubits n satisfies k = O(n/log n). In doing this, we provide an efficient construction of a quantum tensor product expander, which is a generalisation of a quantum expander which in turn generalises classical expanders. We then discuss applications of k-designs. We show that they can be used to improve the efficiency of many existing algorithms and protocols and also find new applications to derandomising large deviation bounds. In particular, we show that many large deviation bound results for Haar random unitaries carry over to k-designs for k = poly(n). In the second part of the thesis, we present some learning and testing algorithms for the Clifford group. We find an optimal algorithm for identifying an unknown Clifford operation. We also give an algorithm to test if an unknown operation is close to a Clifford or far from every Clifford.

Figures (3)

• Figure 2. 1: A diagram showing the different parts of the proof of Lemma \ref{['lem:ApproxUnitaryDesignEquiv']}. The dotted arrows show the correspondence is only for $k=2$. The circled digits refer to the enumerated items below and the factors by the arrows indicate the precision lost in the approximation when converting between the definitions.
• Figure 3. 2: An example of a random circuit. Different lines indicate a different gate is applied at each step.
• Figure 3. 3: The decomposition of the zero chain into $P_1$ and $P_2$. The graph plotted is the zero chain stationary distribution $\pi(x)$.

Theorems & Definitions (214)

• Lemma 1.2.1
• proof
• Lemma 1.2.2
• proof
• Definition 1.2.3
• Definition 1.2.4
• Definition 1.2.5
• Definition 1.2.6
• Definition 1.2.7
• Definition 1.2.8