Source-linked AI summary
A quantum algorithm for evolving open quantum dynamics on quantum computing devices
Zixuan Hu, Rongxin Xia, Sabre Kais
TL;DR
The supplementary analysis develops matrix-based procedures for evolving states and extracting observables in open-system dynamics. It reports agreement with main-text results, while noting that off-diagonal state elements are not directly obtainable without quantum measurements.
Problem
The analysis addresses how to compute final-state populations and observables for open-system dynamics in different bases.
Method
The procedure uses Kraus operators as contractions and evaluates final-state populations through sums of transformed states.
Results
The computed results agree with the smooth curves reported in Figures 2 and 3 of the main text.
Takeaways & Limitations
When the number of operators is small, the quantum algorithm can outperform the classical alternative, with observable extraction generalized to another basis.
Takeaways & Limitations
Off-diagonal elements of the final state cannot be directly obtained without quantum measurements.
Abstract
from arXiv · showhide
Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.
Supplementary information: A quantum algorithm for evolving open quantum dynamics on quantum computing devices
The supplementary information supplies technical proofs, a generalized initial-state algorithm, and the quantum circuits used for demonstrations.
- The supplementary document provides technical details supporting the main-text discussion.It includes proofs, algorithmic modifications, and implementation information.
- Section 1: Section 1 proves that each Kraus operator M_k is a contraction.
- Section 2: Section 2 proves that Õ is a contraction and positive-semidefinite.
- Section 3: Section 3 presents a modified algorithm for an initial state given in general matrix form.
- Section 4: Section 4 lists the quantum circuits used on the IBM Qiskit simulator and IBM Q 5 Tenerife device.
1. Proof that each Kraus operator k M is a contraction
The section proves that every Kraus operator is a contraction by contradiction, using the completeness relation of the operator-sum representation.
- A contraction is an operator that shrinks or preserves the norm of every vector.Its operator norm satisfies sup ||Av||/||v|| ≤ 1.
- The Kraus operators satisfy Σ_k M_k†M_k = I in the operator-sum representation.
- Assuming one Kraus operator is not a contraction yields a vector whose norm increases under that operator.
- The completeness relation bounds each individual contribution by the original vector norm, producing a contradiction.
- Therefore, every Kraus operator M_k is a contraction.
2. Proof that 2
This section establishes that the constructed operator Õ is both contractive and positive-semidefinite, enabling its use in the measurement procedure.
- Õ is constructed from O so that it is a contraction.The proof uses the triangle inequality for the operator norm and the Hilbert-Schmidt norm bound.
- The construction Õ = (I + O†O) is selected because the operator norm of O is harder to calculate.
- The smallest eigenvalue of O is bounded by its Hilbert-Schmidt norm in the positivity argument.
- For any vector v, the eigenvalue bound implies nonnegative quadratic forms for Õ.
- Therefore, Õ is positive-semidefinite.
3. Modified algorithm for the initial state in a general matrix form
The modified algorithm evolves a general initial density matrix by flattening it into a vector, applying unitary dilations of Kraus-related operators, and extracting populations or observables.
- Algorithm construction: The algorithm represents the flattened density matrix as a normalized quantum state and simulates Kraus-operator effects with quantum gates.The left and right Kraus multiplications are converted into Kronecker-product operators.
- Algorithm construction: Each Kraus-related operator is a contraction, so two-dimensional unitary dilations can implement the non-unitary evolution.The dilation acts on an extended space and returns the desired output after projection.
- Gate complexity: The construction uses 2n-dimensional lifted operators and 2^3n-dimensional unitary gates when M_k has dimension n by n.
- Gate complexity: The quantum gate count is 3·2^(3n)+n^2−2^n for each k, versus 3·2^(4n)−2^n classical operations for matrix multiplication.
- Gate complexity: When the number of Kraus operators is small, the quantum algorithm can outperform the classical method by exploiting T's unitarity.
- Physical observables: Projection measurements obtain the diagonal elements, whose sum over k gives the final-state populations in the current basis.
- Physical observables: Off-diagonal elements are carried by the output vectors but cannot be directly obtained without quantum tomography.They are needed for basis changes and observable expectation values.
- Physical observables: Basis transformations require no additional dilations because the corresponding T-based operators are unitary.The additional gates increase the count to 3·4^n for each k.