Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0004029 · The Journal of Chemical Physics

Quantum codes for controlling coherent evolution

Yehuda Sharf, Timothy F. Havel, David G. Cory

Published Dec 22, 2000• 2 citations
arXivHTMLPDF

Abstract

Control over spin dynamics has been obtained in nuclear magnetic resonance (NMR) via coherent averaging, which modifies the effective internal Hamiltonian, and via quantum codes, which can protect against decoherent evolution. Here, we discuss the design and implementation of quantum codes that enable modification of the internal Hamiltonian. A detailed example is given of a quantum code for protecting two data spins from evolution under a weak coupling term in the Hamiltonian, using an “isolated” ancilla that does not evolve on the experimental time scale. The code is realized in a three-spin system by liquid-state NMR spectroscopy on 13C-labeled alanine, and tested for two initial states. It is also shown that with internal interactions and isolated ancillae, codes exist that do not require the ancillae to initially be in a (pseudo-) pure state. Finally, it is shown that even with nonisolated ancillae, quantum codes exist which can protect against evolution under weak coupling. An example is presented for a six-qubit code that protects two data spins to first order.

DOI
10.1063/1.1326071

Related papers

Quantum Codes for Controlling Coherent Evolution

Yehuda Sharf, Timothy F. Havel, David G. Cory

Control over spin dynamics has been obtained in NMR via coherent averaging, which is implemented through a sequence of RF pulses, and via quantum codes which can protect against incoherent evolution. Here, we discuss the design and implementation of quantum codes to protect against coherent evolution. A detailed example is given of a quantum code for protecting two data qubits from evolution under a weak coupling (Ising) term in the Hamiltonian, using an ``isolated'' ancilla which does not evolve on the experimental time scale. The code is realized in a three-spin system by liquid-state NMR spectroscopy on 13C-labelled alanine, and tested for two initial states. It is also shown that for coherent evolution and isolated ancillae, codes exist that do not require the ancillae to initially be in a (pseudo-)pure state. Finally, it is shown that even with non-isolated ancillae quantum codes exist which can protect against evolution under weak coupling. An example is presented for a six qubit code that protects two data spins to first order.

•Apr 7, 2000•2 citations

Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor

Osama Moussa, Jonathan Baugh, Colm A. Ryan +1 more

We report the implementation of a 3-qubit quantum error-correction code on a quantum information processor realized by the magnetic resonance of carbon nuclei in a single crystal of malonic acid. The code corrects for phase errors induced on the qubits due to imperfect decoupling of the magnetic environment represented by nearby spins, as well as unwanted evolution under the internal Hamiltonian. We also experimentally demonstrate sufficiently high-fidelity control to implement two rounds of quantum error correction. This is a demonstration of state-of-the-art control in solid state nuclear magnetic resonance, a leading test bed for the implementation of quantum algorithms.

arXiv:1108.4842•Physical Review Letters•Oct 10, 2011•50 citations

Quantum control of nuclear magnetic resonance spin systems

Jun Li, Jiangyu Cui, Xiaodong Yang +6 more

With the development of quantum information science, the active manipulation of quantum systems is becoming an important research frontier. To build realistic quantum information processors, one of the challenges is to implement arbitrary desired operations with high precision on quantum systems. A large number of quantum control methods and relevant numerical techniques have been put forward in recent years, such as quantum optimal control and quantum feedback control. Nuclear magnetic resonance (NMR) spin systems offer an excellent testbed to develop benchmark tools and techniques for controlling quantum systems. In this review paper, we briefly introduce some of the basic control ideas developed for NMR systems in recent years. We first explain, for the liquid spin systems, the physics of various couplings and the causes of relaxation effects. These mechanisms govern the system dynamics, and thus are crucial for constructing rigorous and efficient control models. We also identify three types of available control means: 1) raido-frequency fields as coherent controls; 2) phase cycling, gradient fields and relaxation effects as non-unitary controls; 3) radiation damping effect as feedback control mechanism. Then, we elucidate some important control tasks, which may arise from the conventional NMR spectroscopy (e.g., pulse design and polarization transfer) or from quantum information science (e.g., algorithmic cooling and pseudo-pure state preparation). In the last part, we review some of the most important control methods that are applicable to NMR control tasks. For systems with a relatively small number of spins, it is possible to use analytic optimal control theory to realize the target unitary operations. However, for larger systems, numerical methods are necessary. The gradient ascent pulse engineering algorithm and pulse compiler techniques are the most successful techniques for implementing complicated quantum networks currently. There are some interesting topics of utilizing radiation damping and relaxation effects to achieve more powerful controls. Finally, we give an outline of the possible future work.

Acta Physica Sinica•Jan 1, 2015•4 citations

A study of quantum error correction by geometric algebra and liquid-state NMR spectroscopy

Yehuda Sharf, David G. Cory, Shyamal S. Somaroo +4 more

Quantum error correcting codes enable the information contained in a quantum state to be protected from decoherence due to external perturbations. Applied to NMR, this procedure does not alter normal relaxation, but rather converts the state of a 'data' spin into multiple quantum coherences involving additional ancilla spins. These multiple quantum coherences relax at differing rates, thus permitting the original state of the data to be approximately reconstructed by mixing them together in an appropriate fashion. This paper describes the operation of a simple, three-bit quantum code in the product operator formalism, and uses geometric algebra methods to obtain the error-corrected decay curve in the presence of arbitrary correlations in the external random fields. These predictions are confirmed in both the totally correlated and uncorrelated cases by liquid-state NMR experiments on 13C-labelled alanine, using gradient-diffusion methods to implement these idealized decoherence models. Quantum error correction in weakly polarized systems requires that the ancilla spins be prepared in a pseudo-pure state relative to the data spin, which entails a loss of signal that exceeds any potential gain through error correction. Nevertheless, this study shows that quantum coding can be used to validate theoretical decoherence mechanisms, and to provide detailed information on correlations in the underlying NMR relaxation dynamics.

arXiv:quant-ph/0004030•Molecular Physics•Sep 1, 2000•26 citations

Use of quadrupolar nuclei for quantum-information processing by nuclear magnetic resonance: Implementation of a quantum algorithm

Ranabir Das, Anil Kumar

Physical implementation of quantum-information processing by liquid-state nuclear magnetic resonance, using weakly coupled spin-$\frac{1}{2}$ nuclei of a molecule, is well established. Nuclei with $\mathrm{spin}>1/2$ oriented in liquid-crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates has been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two developments. First, we implement a quantum algorithm that needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multiqubit gates. We implement the Deutsch-Jozsa algorithm on a spin-$\frac{3}{2}$ (2 qubit) system. The controlled-NOT operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. To the best of our knowledge, this method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-$\frac{1}{2}$ nuclei, the gate time decreases, increasing the clock speed of the quantum computer.

arXiv:quant-ph/0306100•Physical Review A•Sep 10, 2003•54 citations

Automatic Quantum Error Correction

Jeff P. Barnes, Warren S. Warren

Criteria are given by which dissipative evolution in the weak coupling limit can transfer populations and coherences between quantum subspaces, without a loss of coherence. The result is a form of quantum error correction implemented by the joint evolution of a system with a cold bath. It requires no external intervention and, in principle, no ancilla. An example of such a system, consisting of three dipole ordered spin 1/2 particles in a resonator, is given. The qubit, or the triple quantum coherence of the spins, is protected against all spin-flip errors.

arXiv:quant-ph/9912104•Physical Review Letters•Jul 24, 2000•30 citations
Control over spin dynamics has been obtained in nuclear magnetic resonance (NMR) via coherent averaging, which modifies the effective internal Hamiltonian, and via quantum codes, which can protect against decoherent evolution. Here, we discuss the design and implementation of quantum codes that enable modification of the internal Hamiltonian. A detailed example is given of a quantum code for protecting two data spins from evolution under a weak coupling term in the Hamiltonian, using an “isolated” ancilla that does not evolve on the experimental time scale. The code is realized in a three-spin system by liquid-state NMR spectroscopy on 13C-labeled alanine, and tested for two initial states. It is also shown that with internal interactions and isolated ancillae, codes exist that do not require the ancillae to initially be in a (pseudo-) pure state. Finally, it is shown that even with nonisolated ancillae, quantum codes exist which can protect against evolution under weak coupling. An example is presented for a six-qubit code that protects two data spins to first order.
Control over spin dynamics has been obtained in NMR via coherent averaging, which is implemented through a sequence of RF pulses, and via quantum codes which can protect against incoherent evolution. Here, we discuss the design and implementation of quantum codes to protect against coherent evolution. A detailed example is given of a quantum code for protecting two data qubits from evolution under a weak coupling (Ising) term in the Hamiltonian, using an ``isolated'' ancilla which does not evolve on the experimental time scale. The code is realized in a three-spin system by liquid-state NMR spectroscopy on 13C-labelled alanine, and tested for two initial states. It is also shown that for coherent evolution and isolated ancillae, codes exist that do not require the ancillae to initially be in a (pseudo-)pure state. Finally, it is shown that even with non-isolated ancillae quantum codes exist which can protect against evolution under weak coupling. An example is presented for a six qubit code that protects two data spins to first order.
We report the implementation of a 3-qubit quantum error-correction code on a quantum information processor realized by the magnetic resonance of carbon nuclei in a single crystal of malonic acid. The code corrects for phase errors induced on the qubits due to imperfect decoupling of the magnetic environment represented by nearby spins, as well as unwanted evolution under the internal Hamiltonian. We also experimentally demonstrate sufficiently high-fidelity control to implement two rounds of quantum error correction. This is a demonstration of state-of-the-art control in solid state nuclear magnetic resonance, a leading test bed for the implementation of quantum algorithms.
With the development of quantum information science, the active manipulation of quantum systems is becoming an important research frontier. To build realistic quantum information processors, one of the challenges is to implement arbitrary desired operations with high precision on quantum systems. A large number of quantum control methods and relevant numerical techniques have been put forward in recent years, such as quantum optimal control and quantum feedback control. Nuclear magnetic resonance (NMR) spin systems offer an excellent testbed to develop benchmark tools and techniques for controlling quantum systems. In this review paper, we briefly introduce some of the basic control ideas developed for NMR systems in recent years. We first explain, for the liquid spin systems, the physics of various couplings and the causes of relaxation effects. These mechanisms govern the system dynamics, and thus are crucial for constructing rigorous and efficient control models. We also identify three types of available control means: 1) raido-frequency fields as coherent controls; 2) phase cycling, gradient fields and relaxation effects as non-unitary controls; 3) radiation damping effect as feedback control mechanism. Then, we elucidate some important control tasks, which may arise from the conventional NMR spectroscopy (e.g., pulse design and polarization transfer) or from quantum information science (e.g., algorithmic cooling and pseudo-pure state preparation). In the last part, we review some of the most important control methods that are applicable to NMR control tasks. For systems with a relatively small number of spins, it is possible to use analytic optimal control theory to realize the target unitary operations. However, for larger systems, numerical methods are necessary. The gradient ascent pulse engineering algorithm and pulse compiler techniques are the most successful techniques for implementing complicated quantum networks currently. There are some interesting topics of utilizing radiation damping and relaxation effects to achieve more powerful controls. Finally, we give an outline of the possible future work.
Quantum error correcting codes enable the information contained in a quantum state to be protected from decoherence due to external perturbations. Applied to NMR, this procedure does not alter normal relaxation, but rather converts the state of a 'data' spin into multiple quantum coherences involving additional ancilla spins. These multiple quantum coherences relax at differing rates, thus permitting the original state of the data to be approximately reconstructed by mixing them together in an appropriate fashion. This paper describes the operation of a simple, three-bit quantum code in the product operator formalism, and uses geometric algebra methods to obtain the error-corrected decay curve in the presence of arbitrary correlations in the external random fields. These predictions are confirmed in both the totally correlated and uncorrelated cases by liquid-state NMR experiments on 13C-labelled alanine, using gradient-diffusion methods to implement these idealized decoherence models. Quantum error correction in weakly polarized systems requires that the ancilla spins be prepared in a pseudo-pure state relative to the data spin, which entails a loss of signal that exceeds any potential gain through error correction. Nevertheless, this study shows that quantum coding can be used to validate theoretical decoherence mechanisms, and to provide detailed information on correlations in the underlying NMR relaxation dynamics.
Physical implementation of quantum-information processing by liquid-state nuclear magnetic resonance, using weakly coupled spin-12\frac{1}{2}21​ nuclei of a molecule, is well established. Nuclei with spin>1/2\mathrm{spin}>1/2spin>1/2 oriented in liquid-crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates has been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two developments. First, we implement a quantum algorithm that needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multiqubit gates. We implement the Deutsch-Jozsa algorithm on a spin-32\frac{3}{2}23​ (2 qubit) system. The controlled-NOT operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. To the best of our knowledge, this method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-12\frac{1}{2}21​ nuclei, the gate time decreases, increasing the clock speed of the quantum computer.
Criteria are given by which dissipative evolution in the weak coupling limit can transfer populations and coherences between quantum subspaces, without a loss of coherence. The result is a form of quantum error correction implemented by the joint evolution of a system with a cold bath. It requires no external intervention and, in principle, no ancilla. An example of such a system, consisting of three dipole ordered spin 1/2 particles in a resonator, is given. The qubit, or the triple quantum coherence of the spins, is protected against all spin-flip errors.