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Ett QuTiP-baserat ramverk för simulering av brusig Grovers sökalgoritm med kvantfelkorrigering

Loïc Savaton

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

This work develops and tests a simulation framework based on Python and the Quantum Toolbox in Python (QuTiP), a library used to build quantum states, operators, and open-system time evolutions. The objective is to simulate noisy quantum circuits with Lindblad dynamics, a master-equation description in which coherent gate evolution and dissipative noise from the environment are treated to- gether. Grover’s search algorithm is used as a reference circuit. It is not used here to optimize the search algorithm itself, but to study how far a realistic numerical description can be pushed when finite-duration gates, local noise models, auxiliary work qubits called ancillas, encoded states, and recovery procedures are included. The first part of the work constructs a realistic unencoded noisy Grover model. In this model, the oracle, which marks the desired solutions, and the diffuser, which amplifies their probability, are decomposed into elementary gates, and noise acts during the execution of each gate. Three noise mechanisms are compared: dephasing, which destroys phase coherence; amplitude damping, which models relaxation from |1〉 to |0〉, and depolarizing noise, which represents a more symmetric loss of quantum information. Quantum Error Correction (QEC) is then studied as a way to protect logical information by encoding one logical qubit into several physical qubits. The QEC models are analyzed through a hierarchy of benchmarks designed to add realism progressively while keeping the simulations computationally feasible. These bench- marks include ideal recovery, recovery with an explicit time cost, semi-realistic oracle or diffuser dynamics, and the combined semi-realistic case. The results show that QEC can improve the marked-state probability in specific intermediate noise regimes, but that this benefit is strongly limited by recovery time, encoding cost, and the semi-realistic treatment of the Grover blocks. The phase-flip code gives the clearest improvement for dephasing noise, while amplitude-damping codes show more constrained gains, especially when success probability or larger encoded spaces are taken into account. Finally, the QuTiP solvers mesolve and mcsolve are compared. The first directly evolves the density matrix, while the sec- ond uses Monte Carlo wave-function trajectories. The trajectory approach reduces memory pressure but introduces statistical convergence costs. Overall, the project identifies both the physical regimes where QEC is informative and the numerical limits of realistic Python/QuTiP simulations.

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This work develops and tests a simulation framework based on Python and the Quantum Toolbox in Python (QuTiP), a library used to build quantum states, operators, and open-system time evolutions. The objective is to simulate noisy quantum circuits with Lindblad dynamics, a master-equation description in which coherent gate evolution and dissipative noise from the environment are treated to- gether. Grover’s search algorithm is used as a reference circuit. It is not used here to optimize the search algorithm itself, but to study how far a realistic numerical description can be pushed when finite-duration gates, local noise models, auxiliary work qubits called ancillas, encoded states, and recovery procedures are included. The first part of the work constructs a realistic unencoded noisy Grover model. In this model, the oracle, which marks the desired solutions, and the diffuser, which amplifies their probability, are decomposed into elementary gates, and noise acts during the execution of each gate. Three noise mechanisms are compared: dephasing, which destroys phase coherence; amplitude damping, which models relaxation from |1〉 to |0〉, and depolarizing noise, which represents a more symmetric loss of quantum information. Quantum Error Correction (QEC) is then studied as a way to protect logical information by encoding one logical qubit into several physical qubits. The QEC models are analyzed through a hierarchy of benchmarks designed to add realism progressively while keeping the simulations computationally feasible. These bench- marks include ideal recovery, recovery with an explicit time cost, semi-realistic oracle or diffuser dynamics, and the combined semi-realistic case. The results show that QEC can improve the marked-state probability in specific intermediate noise regimes, but that this benefit is strongly limited by recovery time, encoding cost, and the semi-realistic treatment of the Grover blocks. The phase-flip code gives the clearest improvement for dephasing noise, while amplitude-damping codes show more constrained gains, especially when success probability or larger encoded spaces are taken into account. Finally, the QuTiP solvers mesolve and mcsolve are compared. The first directly evolves the density matrix, while the sec- ond uses Monte Carlo wave-function trajectories. The trajectory approach reduces memory pressure but introduces statistical convergence costs. Overall, the project identifies both the physical regimes where QEC is informative and the numerical limits of realistic Python/QuTiP simulations.
The study of the impact of noise on quantum circuits is especially relevant to guide the progress of Noisy Intermediate-Scale Quantum (NISQ) computing. In this paper, we address the pulse-level simulation of noisy quantum circuits with the Quantum Toolbox in Python (QuTiP). We introduce new tools in qutip-qip , QuTiP's quantum information processing package. These tools simulate quantum circuits at the pulse level, leveraging QuTiP's quantum dynamics solvers and control optimization features. We show how quantum circuits can be compiled on simulated processors, with control pulses acting on a target Hamiltonian that describes the unitary evolution of the physical qubits. Various types of noise can be introduced based on the physical model, e.g., by simulating the Lindblad density-matrix dynamics or Monte Carlo quantum trajectories. In particular, the user can define environment-induced decoherence at the processor level and include noise simulation at the level of control pulses. We illustrate how the Deutsch-Jozsa algorithm is compiled and executed on a superconducting-qubit-based processor, on a spin-chain-based processor and using control optimization algorithms. We also show how to easily reproduce experimental results on cross-talk noise in an ion-based processor, and how a Ramsey experiment can be modeled with Lindblad dynamics. Finally, we illustrate how to integrate these features with other software frameworks.
We analyze the modification of entanglement dynamics in the Grover algorithm when the qubits are subjected to single-qubit amplitude-damping or phase-flip noise. We compare quantum trajectories with full density-matrix simulations, analyzing the dynamics of averaged trajectory entanglement (TE) and operator entanglement (OE), in the respective state representation. Although not a genuine entanglement measure, both TE and OE are connected to the efficiency of matrix product state simulations and thus of fundamental interest. As in many quantum algorithms, at the end of the Grover circuit entanglement decreases as the system converges towards the target product state. While we find that this is well captured in the OE dynamics, quantum trajectories rarely follow paths of reducing entanglement. Optimized unraveling schemes can lower TE slightly, however we show that deep in the circuit OE is generally smaller than TE. This implies that matrix product density operator (MPDO) simulations of quantum circuits can in general be more efficient than quantum trajectories. In addition, we investigate the noise-rate scaling of success probabilities for both amplitude-damping and phase-flip noise in Grover's algorithm.
The promise of quantum computing is closer to reality today than ever before, thanks to rapid progress in the development of quantum hardware. Even as qubit lifetimes and gate fidelities continue to improve, realizing robust, fault-tolerant quantum computers is contingent upon the successful implementation of quantum error correction (QEC). Conventional QEC schemes have rather high resource overheads and low threshold requirements, making them challenging to implement on present day hardware. Here, we use a recently developed noise-adapted 3-qubit QEC scheme to demonstrate break-even performance against native amplitude-damping (AD) noise on IBM quantum hardware. We use variational quantum circuits to construct hardware-efficient encoding and decoding circuits. This scheme is probabilistic due to the non-unitary nature of the recovery operators, which are implemented via the block-encoding technique. We demonstrate logical qubit lifetimes exceeding those of the physical qubits by performing multiple rounds of QEC. To further protect the qubits from dephasing due to crosstalk, we incorporate dynamical decoupling into our noise-adapted QEC scheme in a seamless fashion. To account for the post-selection overhead, we define a measure of gain, that allows for faithful performance benchmarking of the protocol. Our analysis suggests that the performance of our protocol is limited primarily by the measurement readout fidelity, and is bound to improve with successive generations of quantum processors.
Abstract This laboratory work presents a practical implementation of Grover’s search algorithm using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is intended to provide hands-on experience with amplitude amplification and oracle-based quantum search. In Laboratory Work No. 7, participants construct Grover’s algorithm step by step. The implementation begins with defining a function that initializes the quantum registers and prepares the initial stage of the circuit. The output qubit associated with the oracle is set to the state ∣1⟩|1\rangle∣1⟩ to enable phase inversion through the phase kickback mechanism. Hadamard gates are then applied to create a uniform superposition over the search space. The oracle is designed to mark the target state by applying a phase shift, effectively flipping the sign of the amplitude corresponding to the solution. This is followed by the diffusion operator (inversion about the mean), which amplifies the probability amplitude of the marked state. The Grover iteration—composed of the oracle and diffusion operator—is repeated an optimal number of times to maximize the probability of measuring the correct result. The laboratory exercise emphasizes the principles of quantum amplitude amplification, circuit modularization, oracle construction, and measurement-based verification. By executing the algorithm on simulators and cloud-accessible quantum hardware, participants observe the quadratic speedup of Grover’s algorithm compared to classical linear search methods. This practical guide bridges theoretical understanding of quantum search with executable quantum circuits, reinforcing the operational aspects of quantum algorithm design within modern quantum computing
Quantum noise is a central challenge in quantum computing across many applications. Extensive work has examined how qubits couple to their environment, leading to decoherence and irreversible relaxation. This work studies a continuous coherent noise model for quantum circuits and compares it with a discrete Pauli model. The focus is on small, coherent gate errors that accumulate across circuit depth. These errors are modeled as random rotations on the Bloch sphere using a von Mises-Fisher distribution. In the small-angle limit, the model reduces to an isotropic Gaussian distribution. We test the model on quantum error-correction circuits based on the [[5,1,3]] and [[7,1,3]] codes. A variant of Grover's search circuit with different qubit counts is also examined. To enable fair comparison, we introduce a model-independent matching scheme. Pauli and continuous noise channels are aligned using the binary entropy at readout. This isolates the effect of noise structure at fixed uncertainty. An approximate analytical method for the propagation of coherent errors is also developed. The method tracks error distributions both on Clifford and non-Clifford circuits without full Monte Carlo sampling. It reduces simulation cost while preserving accuracy for circuit-level error estimates. The approximation is validated against brute-force simulations, identifying its regime of validity with Clifford and non-Clifford circuits under error correction. Our results show that continuous coherent noise and Pauli noise lead to comparable logical-error trends in the stabilizer-code circuits studied here. This behavior is consistent with syndrome extraction projecting a general single-qubit error onto Pauli-error subspaces before recovery. They also show where simplified propagation models work well and where their accuracy is reduced.
Classical database join operations are a significant performance bottleneck in data-intensive applications, typically requiring quadratic time proportional to the size of the input. Quantum search algorithms, such as Grover’s algorithm, promise asymptotic speedups for unstructured search and can be adapted to accelerate equality-based joins. In this paper, we present a systematic simulation study of a Grover-based quantum join primitive and a hybrid classical-quantum join scheme, comparing them with a baseline classical nested-loop join. Parameterised Grover circuits have been implemented for marked-item search, together with resource estimation (qubit count, circuit depth, and gate counts), using the Qiskit Aer simulator. The classical join is modelled as a nested-loop equality join over two arrays, while the hybrid scheme uses a classical Bloom filter to reduce the candidate space, followed by a Grover search over the remaining items. To approximate near term hardware, we incorporate depolarizing gate noise and readout noise via Aer noise models and investigate two mitigation techniques: readout error calibration and zero noise extrapolation. Our simulations, conducted in a Python environment without GPU acceleration, demonstrate that the Grover based and hybrid joins exhibit the expected sub-quadratic scaling in runtime compared to the classical nested-loop join for problem sizes up to N = 215. We additionally observe that realistic noise significantly degrades the raw success probability of the quantum join, while combining readout mitigation with zero noise extrapolation partially restores accuracy. These results provide a resource-aware and noise-aware baseline for evaluating quantum-accelerated joins on future fault-tolerant or high-fidelity noisy intermediate-scale quantum (NISQ) devices.