Session Q50: Quantum Control Theory and Entanglement

Shortcuts to Adiabaticity in Krylov Space

Shortcuts to adiabaticity provide fast protocols for quantum state preparation in which the use of auxiliary counterdiabatic controls circumvents the requirement of slow driving in adiabatic strategies. While their development is well established in simple systems, their engineering and implementation are challenging in many-body quantum systems with many degrees of freedom. We show that the equation for the counterdiabatic term, equivalently the adiabatic gauge potential, is solved by introducing a Krylov basis. The Krylov basis spans the minimal operator subspace in which the dynamics unfolds and provides an efficient way to construct the counterdiabatic term. We apply our strategy to paradigmatic single and many-particle models. The properties of the counterdiabatic term are reflected in the Lanczos coefficients obtained in the course of the construction of the Krylov basis by an algorithmic method. We examine how the expansion in the Krylov basis incorporates many-body interactions in the counterdiabatic term.   

Fast-forward scaling and shortcuts to adiabaticity

Fast-forward scaling is a method to realize given dynamics within a different time, and shortcuts to adiabaticity are methods to realize adiabatic time evolution or to obtain a final result of adiabatic time evolution within a short time. Fast-forward scaling can be applied to adiabatic time evolution, and thus it is regarded as one of the methods of shortcuts to adiabaticity. Relations between different methods of shortcuts to adiabaticity have been discussed.

In this talk, we revisit a relation between fast-forward scaling and counterdiabatic driving, which is one of the methods of shortcuts to adiabaticity. By adopting expression of fast-forward scaling in [Hatomura, SciPost Physics 15, 036 (2023)], we reformulate application of fast-forward scaling to counterdiabatic driving and the procedure of regularization. Then, we find clear equivalence between fast-forward scaling of adiabatic time evolution and counterdiabatic driving. Our finding builds a bridge between fast-forward scaling and counterdiabatic driving.   

Feedback-based quantum algorithm inspired by Counterdiabatic Driving

The feedback-based quantum algorithm is a ``fully quantum" algorithm that uses a quantum control method to design quantum circuits to prepare the ground state of a quantum many-body system as well as solve combinatorial optimization problems. Such a quantum circuit is built iteratively via measurements of operators that determine the quantum control. A feedback-based circuit has a similarity to that of the Quantum approximate optimization algorithm (QAOA), in the sense that the circuit is built using alternate usage of unitaries obtained from the problem Hamiltonian as well as an additional Hamiltonian. Here, we propose that by including one additional control parameter inspired by counterdiabatic driving, one can accelerate the population transfer to the low-energy states within a shorter time scale compared to a single control parameter. We apply our algorithm to various one-dimensional Ising Hamiltonians. We find that the population transfer is much faster with an additional control parameter inspired by counterdiabatic drive.   

Robustness of controlled Hamiltonian approaches to unitary quantum gates

We examine the effectiveness and resilience of achieving quantum gates employing three approaches stemming from quantum control methods: counterdiabatic driving, Floquet engineering, and inverse engineering. We critically analyse their performance in terms of the gate infidelity, the associated resource overhead based on energetic cost, the susceptibility to time-keeping errors, and the degradation under environmental noise. Despite significant differences in the dynamical path taken, we find a broadly consistent behavior across the three approaches in terms of the efficacy of implementing the target gate and the resource overhead. Furthermore, we establish that the functional form of the control fields plays a crucial role in determining how faithfully a gate operation is achieved. Our results are demonstrated for single qubit gates, with particular focus on the Hadamard gate, and we discuss the extension to N-qubit operations.   

Quantum Shortcut to High-Fidelity State Preparation in a Jaynes-Cummings Lattice

We utilize the counter-diabatic driving approach to achieve a quantum shortcut to high-fidelity state preparation in a two-site Jaynes-Cummings (JC) Lattice. Counter-diabatic Hamiltonians can eliminate undesired diabatic transitions during an adiabatic evolution and enable high-fidelity state preparation. However, the practical implementation of such Hamiltonians is often challenging. Here we present a counter-diabatic Hamiltonian for a two-site JC lattice with a single polariton, which can be realized by simply tuning local qubit-cavity couplings in the JC lattice. We then extend this approach to state preparation in a two-site lattice with two polaritons and multi-site lattices with one polariton, where the counter-diabatic Hamiltonian includes non-local couplings. Our numerical result shows that this approach can be robust against errors in the ramping parameters.   

State Preparation in a Jaynes-Cummings Lattice with Quantum Optimal Control: Threshold Time and Quantum Speed Limit.

We develop a quantum optimal control (QOC) algorithm to study state preparation in a finite-sized Jaynes-Cummings (JC) lattice with unit fillings. In our previous study [1], we showed that desired many-body states can be generated with high fidelity when the evolution time is beyond a threshold time that depends on the bounds of the system parameters. In this work, we improve the QOC algorithm to achieve a better understanding of the relation between the threshold time and the quantum speed limit (QSL). With the current algorithm, the infidelity of the prepared states can be lowered to below 10^-5. We also calculate energy fluctuations and entanglement of the prepared states during the time evolution, which lead to more insights on the threshold time. 

 

[1] Parajuli et.al. State Preparation in a Jaynes-Cummings Lattice with Quantum Optimal Control, arXiv: 2306.11968   

New methods for quantum control and circuit synthesis with symmetry-respecting interactions

Universality of 2-qubit unitary transformations is one of the cornerstones of quantum computing, with many applications and implications extending beyond this field. However, it has been shown that this universality does not hold in the presence of global continuous symmetries such as U(1) and SU(2): generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems [I. Marvian, Nature Physics (2022)]. Further investigations have revealed that the restrictions imposed by the locality of interactions vary significantly for different symmetry groups. While there is currently no comprehensive theory for general symmetry groups, recent work has developed the theory of symmetric quantum circuits for the case of Abelian symmetries.  In this talk, first, I will give an overview of this ongoing project. In the second part, I will focus on the special case of energy-conserving unitaries, i.e., those that conserve the sum of Pauli Z operators on all qubits, corresponding to a global U(1) symmetry. I will present explicit circuit synthesis methods for realizing all such unitaries with XY interaction alone, using 2 ancilla qubits. In particular, I will discuss the properties of circuits containing only the square-root-of-iSWAP gates.   

Two-dimensional Automata of Highly Entangled States

We develop an approach to generate highly entangled ground states of local Hamiltonians. We extend the definition of pushdown automata to two dimensions by adding a stack structure to the two-dimensional online tesselation automata. To classify the complexity of the generated states, we consider the generalization of context-free language (CFL) by adapting the Chomsky normal form to two-dimensional geometries. Such generalization enables us to construct local projectors forming local Hamiltonians. Through the extended Chomsky normal form, we also investigate the relation between general language complexity and Hamiltonian locality. Finally, we interpret the normal form as the local transformation of the random processes, the universality of which can help to classify the entanglement scaling.   

Entanglement-Thermodynamical Study of Quantum Energy Teleportation in One-Dimensional Heisenberg Model

We study the quantum energy teleportation in one-dimensional Heisenberg model under the open boundary condition. A local magnetic field is applied at the edge sites to control the degree of the ground-state entanglement. In the teleportation protocol, Alice performs a projective measurement at one edge site, while Bob performs a feedback control at the other edge dependent on the Alice's measurement result to extract energy. We find that the energy extracted by Bob takes a maximum at intermediate value of the local magnetic field. We also find that this magnetic-field behavior is almost proportional to information-entropy changes due to the Bob's feedback control. The role of a feedback control in an entropy change is discussed in term of entanglement thermodynamics.   

An SU(2) symmetric Semidefinite Relaxation for Ground States of Heisenberg Models

The Heisenberg model plays a central role in understanding quantum materials, however, the task of obtaining the ground state of a given Heisenberg-type Hamiltonian is in general very difficult and is known to be QMA-complete. Here, we focus on the fact that for the (antiferromagnetic) Ising model, computational complexity theory provides strong evidence for the Goemans-Williamson algorithm, a semidefinite programming (SDP) based approach, to be *optimal* in the approximation ratio sense (for a particular convention of the base energy choice corresponding to the so-called "Max Cut" problem). We study the "quantum Max Cut" problem, a quantum generalization of this, which corresponds to the Heisenberg model with again a particular convention for the base energy. Following the spirit of valence-bond basis, by using an SU(2) symmetric basis for operators, we construct an SDP algorithm that is both theoretically most natural and practically implementable for the first time. We prove that it could be regarded as a first-order of a systematically improving sequence of approximations (i.e. a properly converging Lasserre/NPA-hierarchy), and furthermore give (in)exactness proofs for some families of graphs. Its potential use for understanding ground states of actual condensed matter stems as well as how it connects to "frustration-free" models known in the context of exact solvability will be discussed.   

Bundled matrix product states represent low-energy excitations faithfully

Finding the ground-state energy of many-body lattice systems is exponentially costly due to the size of the Hilbert space. Ground-state wave functions satisfying the area law of entanglement entropy can be efficiently expressed as a matrix product states (MPS) for local, gapped Hamiltonians. The extension to a bundled MPS describes excitations, and we provide a formal proof of its efficiency. We define a bundled density matrix as a set of independent density matrices which are all written in a common (truncated) basis. We demonstrate that the truncation error is a practical metric that determines how well an excitation is described. We also show that states with volume law entanglement are not necessarily more costly to include in the bundle. The same is true for gapless systems if sufficient lower energy solutions are already present. This result implies that bundled MPSs can describe low-energy excitations without significantly increasing the bond dimension over the cost of the ground-state calculation with caveats we explain.