A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

We construct a random unitary Gaussian circuit for continuous-variable (CV) systems subject to Gaussian measurements. We show that when the measurement rate is nonzero, the steady state entanglement entropy saturates to an area-law scaling. This is different from a many-body qubit system, where a generic entanglement transition is widely expected. Due to the unbounded local Hilbert space, the time scale to destroy entanglement is always much shorter than the one to build it, while a balance could be achieved for a finite local Hilbert space. By the same reasoning, the absence of transition should also hold for other non-unitary Gaussian CV dynamics.

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i)transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii)transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii)projecting this ancilla with near unity success probability.

Quantum simulation is one of the main contents of quantum information science, aiming to simulate and investigate poorly controllable or unobtainable quantum systems by using controllable quantum systems. Quantum simulation can be implemented in quantum computers, quantum simulators, and small quantum devices. Non-Hermitian systems have aroused research interest increasingly in recent two decades. On one hand, non-Hermitian quantum theories can be seen as the complex extensions of the conventional quantum mechanics, and are closely related to open systems and dissipative systems. On the other hand, both quantum systems and classical systems can be constructed as non-Hermitian systems with novel properties, which can be used to improve the precision of precise measurements. However, a non-Hermitian system is more difficult to simulate than a Hermitian system in that the time evolution of it is no longer unitary. In this review, we introduce recent research progress of quantum simulations of non-Hermitian systems. We mainly introduce theoretical researches to simulate typical non-Hermitian quantum systems by using the linear combinations of unitaries, briefly showing the advantages and limitations of each proposal, and we briefly mention other theoretical simulation methods, such as quantum random walk, space embedded and dilation. Moreover, we briefly introduce the experimental quantum simulations of non-Hermitian systems and novel phenomena in nuclear magnetic resonance, quantum optics and photonics, classical systems, etc. The recent progress of the combinations of quantum simulation and non-Hermitian physics has promoted the development of the non-Hermitian theories, experiments and applications, and expand the scope of application of quantum simulations and quantum computers.

If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.

Existing approaches to analog quantum simulations of time-dependent quantum systems rely on perturbative corrections to quantum simulations of time-independent quantum systems. We overcome this restriction to perturbative treatments with an approach based on flow equations and a multimode Fourier expansion. The potential of the quantum simulations that can be achieved with our approach is demonstrated with the pedagogical example of a Λ system and the quench in finite time through a quantum phase transition of a Chern insulator in a driven noninteracting Hubbard system. The example of the Λ system demonstrates the ability of our approach to describe situations beyond the validity of adiabatic approximations. Published by the American Physical Society 2024

Quantum correlation provides a promising measure beyond entanglement. Here, we propose a necessary and sufficient condition for nonzero quantum correlation in continuous variable systems, which is simple and easy to perform in terms of a marker Qr. In order to get this condition, we introduce continuous-variable local orthogonal bases of the operator space, which are generalized from the orthogonal basis sets in local operator space for discrete variables. Based on this, we obtain the marker Qr for all bipartite continuous variable states, and provide several examples including two-mode Gaussian and non-Gaussian states. Our result may provide a candidate for quantum correlation measures, and can be measured by designed quantum circuits.

In this paper we construct a minimal faithful representation of the (2 m+ 2) -dimensional complex general Diamond Lie algebra, Dm(C) , which is isomorphic to a subalgebra of the special linear Lie algebra sl(m+ 2 , C). We also construct a faithful representation of the real general Diamond Lie algebra Dm which is isomorphic to a subalgebra of the special symplectic Lie algebra sp(2 m+ 2 , R). Furthermore, we describe Leibniz algebras with corresponding (2 m+ 2) -dimensional real general Diamond Lie algebra Dm and such that the ideal generated by the squares of elements provides a faithful representation of Dm.

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

Richard Feynman's observation that certain quantum mechanical effects cannot be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used these quantum effects. This speculation proved justified when Peter Shor described a polynomial time quantum algorithm for factoring intergers. In quantum systems, the computational space increases exponentially with the size of the system, which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new nontraditional programming techniques. The aim of this paper is to guide computer scientists through the barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to exploiting the power of quantum parallelism are explained. We conclude with a discussion of quantum error correction.

We propose a formalism of quantum subsystems for reduced descriptions of quantum systems. This unifies a number of well-known methods and less common approaches. The main mathematical ingredients are completely positive maps and correlation functions. In this formalism generalized quantum systems can be composed and there is a notion of generalized entanglement. Models of fermionic and bosonic systems and also quantum systems described by the SU(2) symmetry are studied.

We present a scheme to realize quantum thermal transistor effects in a continuous-variable electromechanical system including two microwave cavities and one mechanical oscillator. The thermal noise fluxes between the quantum system and its baths are evaluated by quantum master equation. It is shown that the thermal noise flux at one microwave cavity as an emitter can be dissipated into the other as a collector by combining the heating Stokes and cooling anti-Stokes processes. The indirect energy transfers between the two microwave modes can be significantly amplified by small energy changes at the mechanical oscillator as the base. The extremely high amplification depends sensitively on the detunings of the two microwave modes, which provides a new tool for precision measurements. This study opens the door for constructing quantum thermal transistors using various continuous-variable systems and is well accessible based on current experimental techniques.

Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area. In this work, we construct optimized product formulas of orders 3–6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previously published alternatives. They can also be used as basic methods in recursive procedures to increase the order of approximation. We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.

Quantum computers perform computations exploiting quantum mechanics to a possible advantage, allowing us to prepare and manipulate states that do not have a classical equivalent. In particular, phenomena like superposition and entanglement may enable quantum computers to outperform their classical counterparts in certain applications. Implementing these useful quantum algorithms is contingent upon building accurate quantum hardware that is not affected by noise. Environmental noise decreases coherence time of qubits, meaning that qubits do not stay in a desired state long enough to carry out a complex computation. To that end, harnessing the full power of quantum computers necessitate characterization of noise sources and how they impact a given quantum system. Often times, T1and T2 are used to quantify noise. In this project, we provide an approach as to how T1 and T2 values are calculated and simulated for quantum systems. In addition, we compare simulated values of T1 and T2 with those of real quantum computer’s measurements. IBMQ Experience, an open source software allowing users to simulate and use real quantum hardware, is used. QuTip, a Python-based toolbox offering quantum simulation tools for open quantum systems, is also used.

Thermodynamic resources, beyond their well-known usefulness in work extraction and other thermodynamic tasks, are often important also in tasks that are not evidently thermodynamic. Here we develop a framework for identifying such resources in diverse applications of bosonic continuous-variable systems. Introducing the class of bosonic linear thermal operations to model operationally feasible processes, we apply this model to identify uniquely quantum properties of bosonic states that refine classical notions of thermodynamic resourcefulness. Among these are (1) a suite of temperature-like quantities generalizing the equilibrium temperature to quantum, non-equilibrium scenarios; (2) signal-to-noise ratios quantifying a system’s capacity to carry information in phase-space displacement; and (3) well-established non-classicality measures quantifying the resolution in sensing and parameter estimation tasks.

Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mover accent="true"> <a:mi>H</a:mi> <a:mo>̂</a:mo> </a:mover> <a:mrow> <a:mo>(</a:mo> <a:mi>t</a:mi> <a:mo>)</a:mo> </a:mrow> <a:mo>=</a:mo> <a:msubsup> <a:mo>∑</a:mo> <a:mrow> <a:mi>l</a:mi> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:mrow> <a:mi>L</a:mi> </a:msubsup> <a:msub> <a:mi>η</a:mi> <a:mi>l</a:mi> </a:msub> <a:mrow> <a:mo>(</a:mo> <a:mi>t</a:mi> <a:mo>)</a:mo> </a:mrow> <a:msub> <a:mover accent="true"> <a:mi>g</a:mi> <a:mo>̂</a:mo> </a:mover> <a:mi>l</a:mi> </a:msub> </a:mrow> </a:math> . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mover accent="true"> <d:mi>U</d:mi> <d:mo>̂</d:mo> </d:mover> <d:mrow> <d:mo>(</d:mo> <d:mi>t</d:mi> <d:mo>)</d:mo> </d:mrow> <d:mo>=</d:mo> <d:msubsup> <d:mo>∏</d:mo> <d:mrow> <d:mi>l</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> </d:mrow> <d:mi>L</d:mi> </d:msubsup> <d:msup> <d:mi>e</d:mi> <d:mrow> <d:msub> <d:mi mathvariant="normal">Λ</d:mi> <d:mi>l</d:mi> </d:msub> <d:mrow> <d:mo>(</d:mo> <d:mi>t</d:mi> <d:mo>)</d:mo> </d:mrow> <d:msub> <d:mover accent="true"> <d:mi>g</d:mi> <d:mo>̂</d:mo> </d:mover> <d:mi>l</d:mi> </d:msub> </d:mrow> </d:msup> </d:mrow> </d:math> . By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi mathvariant="normal">Λ</h:mi> <h:mi>l</h:mi> </h:msub> <h:mrow> <h:mo>(</h:mo> <h:mi>t</h:mi> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> from the Hamiltonian ones <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>η</j:mi> <j:mi>l</j:mi> </j:msub> <j:mrow> <j:mo>(</j:mo> <j:mi>t</j:mi> <j:mo>)</j:mo> </j:mrow> </j:mrow> </j:math> . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>L</k:mi> <k:mo>≥</k:mo> <k:mn>6</k:mn> </k:mrow> </k:math> ). In this work, we introduce , a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi mathvariant="fraktur">su</l:mi> <l:mo>(</l:mo> <l:mn>1</l:mn> <l:mo>,</l:mo> <l:mn>1</l:mn> <l:mo>)</l:mo> </l:mrow> <l:mo>,</l:mo> <l:mo> </l:mo> <l:mrow> <l:mi mathvariant="fraktur">su</l:mi> <l:mo>(</l:mo> <l:mn>2</l:mn> <l:mo>)</l:mo> </l:mrow> <l:mo>,</l:mo> <l:mo> </l:mo> <l:mrow> <l:mi mathvariant="fraktur">sl</l:mi> <l:mo>(</l:mo> <l:mn>2</l:mn> <l:mo>)</l:mo> </l:mrow> </l:math> , and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi mathvariant="fraktur">so</p:mi> <p:mo>(</p:mo> <p:mn>2</p:mn> <p:mo>,</p:mo> <p:mn>1</p:mn> <p:mo>)</p:mo> </p:mrow> </p:math> , and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mrow> <r:mi mathvariant="italic">SU</r:mi> </r:mrow> <r:mrow> <r:mo>(</r:mo> <r:mi>N</r:mi> <r:mo>)</r:mo> </r:mrow> </r:math> using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi mathvariant="fraktur">su</t:mi> <t:mo>(</t:mo> <t:mn>2</t:mn> <t:mo>)</t:mo> </t:mrow> <t:mo>,</t:mo> <t:mo> </t:mo> <t:mrow> <t:mi mathvariant="fraktur">su</t:mi> <t:mo>(</t:mo> <t:mn>3</t:mn> <t:mo>)</t:mo> </t:mrow> </t:math> , and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mrow> <w:mi mathvariant="fraktur">su</w:mi> <w:mo>(</w:mo> <w:mn>4</w:mn> <w:mo>)</w:mo> </w:mrow> </w:math> , including the explicit construction of a set of quantum gates for universal quantum computation.