Recent Research Highlight

Noise-resilient phase estimation with randomized compiling

Quantum phase estimation is a key step in many quantum algorithms, such as Shor's algorithm for factorization, the HHL algorithm for solving systems of linear equations, and the estimation of Hamiltonian energy spectra. However, phase estimation algorithms based on quantum Fourier transform require many auxiliary qubits and need quantum error correction to resist noise. Both of these requirements are difficult to achieve in the current era of noisy intermediate-scale (NISQ) quantum computing. In order to reduce qubit consumption, a phase estimation that does not require a controlled unitary operator was proposed by S. Kimmel, G. H. Low, and T. J. Yoder. However, this algorithm is still affected by noise during actual operation. To get meaningful results, it is necessary to develop corresponding error mitigation techniques.

We developed a novel error mitigation method tailored for control-free phase estimation in quantum computing. We have confirmed through theoretical proof that the phases of a unitary operator are immune to noise channels, provided the channels have only Hermitian Kraus operators under first-order correction. This revelation enables the identification of certain benign types of noise specifically for phase estimation. By additionally incorporating a randomized compiling protocol [Proposed by J. J. Wallman and J. Emerson], we can effectively transformed generic noise in the phase estimation circuits into stochastic Pauli noise. The significance of this transformation is that it aligns with the conditions of their theorem, thereby facilitating a noise-resilient phase estimation without any quantum resource overhead.

Related Publications: Yanwu Gu, Yunheng Ma, Nicolo Forcellini, Dong E. Liu, Phys. Rev. Lett. 130, 250601 (2023)

Channel Spectrum Benchmarking

Noise is the primary hindrance to the evolution of scalable quantum computation. Understanding and mitigating this noise through quantum benchmarking is crucial for developing advanced quantum processors. The current benchmarking protocols have had their limitations, mainly restricted to a particular set of quantum gates or unable to describe the individual target gate’s performance accurately. However, current benchmarking methods are either limited to a specific subset of quantum gates or cannot directly describe the performance of the individual target gate. To overcome these limitations, we propose channel spectrum benchmarking (CSB), a method to infer the noise properties of the target gate, including process fidelity, stochastic fidelity, and some unitary parameters, from the eigenvalues of its noisy channel.

CSB allows for the precise analysis of process fidelity, stochastic fidelity, and various unitary parameters by examining the eigenvalues of a gate's noisy channel. Distinctively, CSB can benchmark a wide array of universal gates and is scalable to multiple qubit systems, making it a versatile tool for various quantum computing applications. Its methodology, involving a specially chosen initial state and control-free phase estimation circuits, ensures simplified data processing and accurate results, ultimately aiding in the development of robust and advanced quantum processors. 

Related Publications: Yanwu Gu, Wei-Feng Zhuang, Xudan Chai, Dong E. Liu,  Nature Communications  14, 5880 (2023)

Coherent Deviations and error threshold theorem in realistic quantum error corrections

A key concept of fault tolerance and QEC is the “error threshold theorem”, which states that if the physical error rate is below an error threshold, quantum computation with arbitrary logical accuracy can be implemented in the noisy quantum devices. Although enjoying widespread acceptance within the quantum computing community, this theorem can only be considered well-established if the quantum devices are solely susceptible to independent stochastic noise of a classical nature. However, actual quantum devices suffer from imperfect calibration and control in gate operations, causing quantum deviation or coherent noise, which is a more practical type of noise. On the other hand, understanding this quantum or coherent error in QEC is exceedingly challenging due to the lack of analytical and numerical tools. This motivates us to develop a theoretical framework to study the elusive and unavoidable coherent error problems in QEC and establish a more practical error threshold theorem.

Here, we consider topological surface code, with both stochastic Pauli noise and coherent noise on the multi-qubit entanglement gates during both code space preparation and stabilizer checks. We map a multi-round error detection protocol to a 3D  lattice gauge theory coupled to a 2D lattice gauge model [1]. Therefore, for the first time, we provide a theoretical tool to study both the coherent and the stochastic error problems for QEC and threshold theorem. Remarkably, our findings indicate that in the limit of large code distance [Middle Figure], imperfect initial state preparation may result in the failure of QEC even below a theoretical threshold, contradicting the commonly held notion that logical errors should be completely suppressed [Left Figure]. What is not excessively frustrating is that: With a finite code size , the effectiveness of the pragmatic QEC remains viable when the error rate associated with the state preparation is below a size-dependent crossover scale  [Right Figure]. For the first time, we obtain a practical error threshold theorem which do not follow the commonly held notion, notably identifying that coherent deviation in the initial QEC state preparations could cause a critical problem.

Related Publications: Yuanchen Zhao, Dong E. Liu, "Lattice gauge theory and topological quantum error correction with quantum deviations in the state preparation and error detection", PNAS Nexus 4, pgaf063  (2025)  Link 

Unified Theoretical Framework connecting Measurements and Disorders

In recent years, there has been a growing fascination with many-body systems and random circuits subjected to measurements, commonly referred to as monitored quantum systems. Notably, considerable attention has been devoted to investigating the phenomenon of subsystem entanglement transition within such systems. Nonetheless, the existing body of research primarily centers on this specific aspect, while other properties of monitored quantum systems remain largely unexplored. Consequently, the comprehensive understanding of these systems and the identification of potential applications hinge upon unraveling the currently uncharted characteristics. Moreover, a significant gap persists in establishing the intrinsic relationship between entanglement transitions induced by measurements and those arising from disorders, despite the resemblances exhibited by both phenomena.

The unification of measurements and disorders, seemingly distinct concepts, is achieved through the adoption of the Keldysh nonlinear sigma model as a theoretical framework. This unexpected and pioneering unification, elucidated by the functional Keldysh field theory, represents a significant advancement in the field. By incorporating the perspectives and methodologies of disordered systems, a well-established research domain since Anderson's seminal work, we are empowered to investigate monitored systems more comprehensively.

Related Publications: Qinghong Yang, Yi Zuo, Dong E. Liu, "Keldysh Nonlinear Sigma Model for a Free-Fermion Gas under Unconditional Continuous Measurements", Phys. Rev. Research 5, 033174 (2023) Link 

Dissipative Floquet Majorana Modes in Proximity-Induced Topological Superconductors

Floquet engineering manipulates quantum systems using periodic driving, holding promise for realizing topological nontrivial band structures and exotic quantum states. However, system-bath couplings introduce complex statistical behaviors in open Floquet systems, necessitating self-consistent treatments and consideration of realistic conditions to comprehend such elusive nonequilibrium systems. For instance, periodically driven topological superconductors, such as Floquet Majorana modes, typically rely on proximity-induced effects. However, under external driving, it becomes essential to treat the superconductor as an external bath, contributing both Cooper pair tunneling and dissipative effects, leading to nonequilibrium steady states. Those concerns raise questions about the validity of the Floquet picture under interplay between nonequilibrium conditions and strong dissipations.

We study a realistic Floquet topological superconductor, a periodically driven nanowire proximitized to an equilibrium s-wave superconductor. In particular, we find that the Floquet Majorana zero and π modes become dissipative, and can no longer be simply described by the Floquet topological band theory. We also propose an effective model to simplify the calculation of the lifetime of these Floquet Majoranas and find that the lifetime can be engineered by the external driving field.

Related Publications: Zhesen Yang, Qinghong Yang, Jiangping Hu, Dong E. Liu, Phys. Rev. Lett. 126, 086801 (2021)