Efficient Unitary T-designs from Random Sums (2402.09335v1)

Abstract: Unitary $T$-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary $T$-designs for $n$-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate $T$-designs in the diamond norm using $O(T^{5+o(1)} n^2)$ quantum gates. In this work, we provide a new construction of $T$-designs via random matrix theory using $\tilde{O}(T^2 n^2)$ quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary $T$-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.

• The paper introduces a novel construction using random matrix theory that significantly reduces the quantum gate count required for unitary T-design generation.
• It approximates the Haar measure by employing i.i.d. random Hermitian matrices to simulate the Gaussian Unitary Ensemble with exponential efficiency improvements.
• The approach connects quantum query complexity with random matrix ensembles, paving the way for efficient Hamiltonian simulation and scalable quantum computing.

Overview of Efficient Unitary $T$-Designs from Random Sums

The paper "Efficient Unitary $T$-designs from Random Sums" presents a novel construction for unitary $T$-designs leveraging ideas from random matrix theory and quantum information. Unitary $T$-designs are ensembles of unitary matrices that reproduce the first $T$ moments of the Haar measure, playing a crucial role in various quantum information tasks, including quantum algorithms, benchmarking, and cryptography. This work introduces an efficient method to generate such designs using $\tilde{O}(T^2 n^2)$ quantum gates, improving the existing known constructions that require $\CO(T^{5+o(1)} n^2)$ gates.

Key Contributions

The authors present a construction that significantly reduces the gate complexity for generating unitary $T$-designs:

1. New Construction via Random Matrix Theory: The paper leverages random sums of matrices to approximate the Gaussian Unitary Ensemble (GUE) through independently and identically distributed (i.i.d.) sums of random Hermitian matrices. With this approach, the paper demonstrates that the product of two exponentiated GUE matrices approximates Haar randomness, significantly reducing the gate count needed for an efficient design.
2. Connection with Quantum Query Complexity: Utilizing the polynomial method in quantum query complexity, the authors provide exponentially improved bounds on high moments of certain random matrix ensembles. This offers a novel framework connecting large-dimensional expansions in random matrix theory to quantum query complexity methodologies.
3. Approximation of GUE: The construction approximates the GUE by exploiting sums of random matrices that match the first few moments of GUE, applying a matrix central limit theorem to argue convergence.
4. Implementation and Applications: The work develops efficient Hamiltonian simulation techniques and discusses practical implications for scalable quantum computing, illustrating potential applications in areas like quantum benchmarking and algorithm development.

Numerical Results and Theoretical Implications

The paper robustly supports its claims with quantitative results, demonstrating that the proposed method achieves $\epsilon$-approximation to the Haar distribution exponentially faster than previous constructions for large $n$ and $T$. The main theorem indicates that an $\epsilon$-approximate $T$-design can be achieved, marking a substantial theoretical advancement in our understanding of random quantum circuits and unitary dynamics.

Future Directions

The research opens the door to several directions for further exploration:

• Enhanced Polynomial Method Techniques: Extending the polynomial method to other matrices and ensembles could unveil broader applications in quantum systems.
• Practical Implementations: While the construction is theoretically efficient, exploring its actual implementation on quantum hardware remains an open challenge to address scalability and error tolerance in practical settings.
• Connections to Quantum Chaos and Gravity: The techniques might offer insights into quantum chaotic systems and holography, drawing parallels between complexity growth in quantum gravity and unitary designs.

In conclusion, the paper's innovative approach to unitary $T$-design construction not only transcends previously held constraints in gate complexity but also enriches the fundamental interplay between random matrix theory and quantum computational techniques. This work sets a new precedent for efficient quantum resource allocation in algorithmic designs, promising advancements in both theoretical and applied quantum physics.