YMSC-BIMSA Quantum Information Seminar

The Quantum State Preparation problem aims to prepare an \(n\)-qubit quantum state \(|\psi\_v\rangle =\sum\_{k=0}^{2^n-1}v_k|k\rangle\) from {the} initial state \(|0\rangle^{\otimes n}\), for a given unit vector \(v=(v_0,v_1,v\_2,\ldots,v\_{2^n-1})^T\in \mathbb{C}^{2^n}\) with \(\|v\|_2 = 1\). The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open when ancillary qubits are available. In this talk, we study quantum circuits when there are \(m\) ancillary qubits available. We construct, for any \(m\), circuits that can prepare \(|\psi_v\rangle\) in depth \(\tilde O\big(\frac{2^n}{m+n}+n\big)\) and size \(O(2^n)\), achieving the optimal value for both measures simultaneously. These results also imply a depth complexity of \(\Theta\big(\frac{4^n}{m+n}\big)\) for quantum circuits implementing a general \(n\)-qubit unitary for any \(m \le O(2^n/n)\) number of ancillary qubits. This resolves the depth complexity for circuits without ancillary qubits. And for circuits with exponentially many ancillary qubits, our result quadratically improves the currently best upper bound of {\(O(4^n)\)} to \(\tilde \Theta(2^n)\).

Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide parameter regime. The results can be viewed as (optimal) time-space trade-off bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take.

Vertex operator algebras (VOAs) are a mathematically rigorous formulation of 2d conformal field theory (CFT). In 2008, Yi-Zhi Huang proved a significant result that the representation category of a “rational” and “C_2 cofinite” VOA V (plus some minor assumptions) is a modular tensor category. Huang’s proof relies on a deep understanding of the geometric properties of conformal blocks (a crucial concept in VOA) associated to compact Rieman surfaces of genera 0 and 1. In recent years, these geometric properties have also been proved for higher genus conformal blocks associated to C_2 cofinite and rational VOAs.

However, much less was known for C_2 cofinite but non-rational VOAs. Indeed, only the genus-0 story was clear, which led to Huang-Lepowsky-Zhang’s construction of braided tensor categories for the representations of C_2 cofinite VOAs. The categories are not necessarily semisimple since V is not assumed to be rational. In this talk, we present an ongoing project that gives a systematic approach to the geometric properties of conformal blocks of C_2 cofinite but non-rational VOAs, associated to compact Riemann surfaces of all genera. Our approach generalizes simultaneously Huang-Lepowsky-Zhang’s tensor category theory and higher level Zhu’s algebras.

For the next six weeks, Kishor is giving a series of tutorials on the topic of Quantum Error Correction, which is a topic that will become increasingly important as people start to realize the limitations of NISQ algorithms.

Content of the six tutorials:

1. Classical error correction, basics of quantum error correction and the stabilizer formalism (7 March)

2. Toric code (14 March)

3. Introduction to topology (21 March)

4. Topological quantum codes, Bosonic codes, subsystem codes (28 March)

5. Approximate quantum error correction, Fault tolerance, Decoders (4 April)

6. Decoders continued, Connections with many body physics, theoretical computer science and black holes (11 April)

We consider the algebraic properties of quantum states represented as polynomials that map binary tuples to complex numbers. Using string diagrams, we review the general factorization results previously established in joint work with Clark, and Jaksch, and explore ongoing developments related to non-orthogonal factorizations based on graded algebra expansions. By imposing a real-valued codomain restriction on these functions, penalty functions are generated that correspond to generalized Ising-type models. We recall the concept of a parent Hamiltonian, a non-negative operator that has a specific state in its zero energy sector, and explain the algebra necessary to control the kernel structure of non-negative pseudo-Boolean functions that represent parent Hamiltonians of quantum states. Our results contribute to the ongoing development of mathematical tools for analyzing and manipulating quantum states, with applications in quantum information science.

Biography:

Dr. Biamonte’s research career has been dedicated to studying the theoretical aspects of quantum information processing, specifically at the interface between quantum applications and physical hardware constraints.

He has held positions as a quantum applications developer at D-Wave Systems Inc. in Vancouver, Canada, and as a Fellow at Harvard University in the Aspuru-Guzik Group. In 2010, Dr. Biamonte obtained his PhD from the University of Oxford. He then worked as a postdoctoral researcher with John Carlos Baez as part of a joint Oxford/Singapore program and as a Lecturer at St Peter’s College Oxford before joining the Institute for Scientific Interchange (ISI Foundation) in Torino, Italy as their Quantum Science Group Leader from 2012-2017.

In 2017, Dr. Biamonte joined the Skolkovo Institute of Science and Technology as a Tenure Track Associate Professor. He was subsequently promoted to Head of the Laboratory for Quantum Information Processing and then to tenured Full Professor. Dr. Biamonte was the first American-born scientist to have successfully defended a higher Doctorate at the Moscow Institute of Physics and Technology. Unfortunately, his time in Moscow was cut short due to the war against Ukraine. Dr. Biamonte currently holds the Jingdong Foundation Chair at the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications.

Dr. Biamonte’s research vision is to develop a data-driven approach to quantify emergent and collective effects in quantum information processing. He is also a dedicated educator, teaching applied and engineering mathematics, quantum information theory, and quantum computation.

Nonlinear dynamics play a prominent role in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. We also established a lower bound showing that nonlinear differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics.

Furthermore, we designed end-to-end quantum machine learning algorithms, combining efficient quantum (stochastic) gradient descent with sparse state preparation and sparse state tomography. We benchmarked instances of training sparse ResNet up to 103 million parameters, and identify the dissipative and sparse regime at the early phase of fine-tuning could receive quantum enhancement. Our work showed that fault-tolerant quantum algorithms could potentially contribute to the scalability and sustainability of most state-of-the-art, large-scale machine learning models.

References:

[1] Liu et al. Efficient quantum algorithm for dissipative nonlinear differential equations, Proceedings of the National Academy of Science 118, 35 (2021), arXiv:2011.03185.

[2] Liu et al. Towards provably efficient quantum