Abstract In Xin et al (2017 Phys. Rev. A 96 062303) and Wei et al (2018 Sci. China Phys. Mech. Astron. 61 70311), the authors reported an algorithm to simulate, in a circuit-based quantum computer, a general quantum channel (QC). However, the application of their algorithm is limited because it entails the solution of intricate non-linear systems of equations in order to obtain the quantum circuit to be implemented for the simulation. Motivated by this issue, in this article we identify and discuss a simple way to implement the simulation of QCs on any d -level quantum system through quantum state preparation algorithms, that have received much attention in the quantum information science literature lately. We exemplify the versatility of our protocol applying it to most well known qubit QCs, to some qudit QCs, and to simulate the effect of Lorentz transformations on spin states. We also regard the application of our protocol for initial mixed states. Most of the given application examples are demonstrated using IBM’s quantum computers.
Abstract In Xin et al (2017 Phys. Rev. A 96 062303) and Wei et al (2018 Sci. China Phys. Mech. Astron. 61 70311), the authors reported an algorithm to simulate, in a circuit-based quantum computer, a general quantum channel (QC). However, the application of their algorithm is limited because it entails the solution of intricate non-linear systems of equations in order to obtain the quantum circuit to be implemented for the simulation. Motivated by this issue, in this article we identify and discuss a simple way to implement the simulation of QCs on any d -level quantum system through quantum state preparation algorithms, that have received much attention in the quantum information science literature lately. We exemplify the versatility of our protocol applying it to most well known qubit QCs, to some qudit QCs, and to simulate the effect of Lorentz transformations on spin states. We also regard the application of our protocol for initial mixed states. Most of the given application examples are demonstrated using IBM’s quantum computers.
In Phys. Rev. A 100, 062317 (2019), the authors reported an algorithm to implement, in a circuit-based quantum computer, a general quantum measurement (GQM) of a two-level quantum system, a qubit. Even though their algorithm seems right, its application involves the solution of an intricate nonlinear system of equations to obtain the angles determining the quantum circuit to be implemented for the simulation. In this article, we identify and discuss a simple way to circumvent this issue and implement GQMs on any d-level quantum system through quantum state preparation algorithms. Using some examples for one qubit, one qutrit, and two qubits, we illustrate the easy of application of our protocol. In addition, we show how one can utilize our protocol for simulating quantum instruments, for which we also give an example. All our examples are demonstrated using IBM's quantum processors.
This work presents a new protocol for simulating quantum systems based on quantum state preparation, with various applications such as the simulation of quantum channels and the implementation of general measurements. In general, these applications are modeled by quantum operations and involve completely positive trace-preserving (CPTP) maps. These maps correspond to sets of operators known as Kraus operators, when applied to the simulation of quantum channels, and general measurement operators, when applied to the implementation of non-selective measurements. In the literature, especially when the experimental platform is circuit-based quantum computing, these operators are employed to adjust the parameters of unitary matrices, which are the ones that actually operate in the experiment. To do this, it is necessary to solve a system of nonlinear equations numerically. The new protocol avoids these complications by applying the operators directly over the analytical description of the quantum systems to be simulated. Through the use of isometries and purifications, it models the quantum operations of interest to the experimentalist, whether they are quantum channels or general measurements, enabling the use of software libraries for symbolic algebra to perform exact and automated manipulation of the state vector, with complex amplitudes, to be prepared for simulation. The algebraic manipulation of the state vector provides greater flexibility in experimental planning, as it can be used on any quantum simulation platform. More specifically, it is not restricted to circuit-based quantum computing. Finally, the use of the new protocol is illustrated in the simulation of one-qubit channels, qudits, Lorentz transformations, mixed-state simulations, POVM implementation, and the certification of quantum contextuality via POVMs.
Abstract The preparation of quantum states is a fundamental subroutine for a broad class of quantum information protocols and is critical for both quantum communication and quantum computation. Building upon the quantum algorithms introduced in previous works [M. B. Pozzobom and J. Maziero, Quantum Inf. Process. 18, 142 (2019)] and [E. R. Gårding et al., Entropy 23, 797 (2021)], the authors of [F. Shahbeigi, M. Karimi and V. Karimipour, Phys. Scr. 97, 025101 (2022)] demonstrated the capability to prepare mixed two-qubit X-real states on quantum computers by extending the methodology originally devised for mixed two-qubit Bell-diagonal states. In this article, we delve into an overlooked pattern within these quantum circuits, allowing us to present a modular algorithm for the preparation of general d -dimensional mixed quantum states using quantum information processors. Our general algorithm has a modular structure, encompassing eigenvalue encoding, entropy injection, and eigenvector preparation. To validate our algorithm, we conduct tests on quantum computers utilizing both X- and non-X-states for two mixed-state qubits, two-ququart Bell-diagonal states, as well as arbitrary random density matrices spanning one, two, and three qubits.
Quantum computers still suffer from errors that keep them in the category of Noisy Intermediate-Scale Quantum (NISQ) computers. Among the various tools capable of circumventing some of the problems of NISQ computers, there is the Variational Quantum Algorithm (VQA). This algorithm can be useful in almost all sub-areas of quantum computing, such as error correction, compilation, combinatorial optimization, and dynamic simulations. These algorithms involve optimization processes in parametric rotation gates and can make use of well-founded optimizers in classical machine learning in computing. Some studies in the formalism of quantum mechanics lead us to possible experiments on a quantum computer, but due to its sensitivity, these experiments require daily calibration. When dealing with an external environment, the system behaves as an open system, governed by a unitary operator in the global state space and described by a completely positive trace-preserving map (CPTP) in the local state space. We call these processes quantum channels, which result in an evolved state over time. Through the formalism of the representation in terms of Kraus operators, we obtain the evolved states through analytical calculations and compare the coherence of this state with the coherence of the evolved state obtained through the protocol of simulating noisy quantum channels proposed in [M. S. Zanetti, D. F. Pinto, M. L. W. Basso, and J. Maziero, Simulating noisy quantum channels via quantum state preparation algorithms, J. Phys. B: At. Mol. Opt. Phys. 56, 115501 (2023)]. During a step of this protocol, state preparations are performed using functions from the Qiskit library. The present work performs state preparations using VQA. We simulate channels associated with bit-flip, phase-flip, bit-phase-flip, depolarizing, amplitude damping, phase damping, generalized amplitude damping, Lorentz, and Heisenberg-Weyl dephasing. The results of simulations with VQA are satisfactory, with the coherence of the simulated state approaching the analytical value. This demonstrates the usefulness of VQA in state preparation for this problem.
Quantum state preparation is an important subroutine for quantum computing. We show that any n-qubit quantum state can be prepared with a Θ(n)-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with d⩾2 nonzero entries, we can reduce the circuit depth to Θ(log(nd)) with O(ndlogd) ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.
Quantum channels describe general transformations of the state of a physical system and play a fundamental role in many fields. It promises wide range of applications, because any physical process can be represented as a quantum channel transforming an initial state into a final state. The evolution of system undergoing a quantum channel is not necessary unitary which makes it unrealisable without ancillary system. Inspired by the method performing non-unitary operator by the linear combination of unitary operations, we proposed a quantum algorithm for the simulation of universal single-qubit channel, described by a convex combination of two generalized extreme channels corresponding to four Kraus operators, and is scalable to arbitrary higher dimension. We demonstrate the whole algorithm experimentally using the universal IBM cloud quantum computer and study properties of different qubit quantum channels. We illustrate the quantum capacity of the general qubit quantum channels, which quantifies the amount of quantum information that can be protected. There is a general agreement between the theoretical predictions and the experiments, which strongly supported our method. By realizing arbitrary qubit channel, this work provides a universal way to explore various properties of quantum channel and novel prospect of quantum communication.