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A Hamiltonian for Quantum Copying

Dima Mozyrsky, Vladimir Privman, Mark Hillery

Published Sep 24, 1996• 4 citations
OpenAlex

Abstract

We derive an explicit Hamiltonian for copying the basis up and down states of a quantum two-state system—a qubit—onto n “copy ” qubits (n ≥ 1) initially all prepared in the down state. In terms of spin components, for spin- 1 2 particle spin states, the resulting Hamiltonian involves n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a quantum-computing controlled-NOT gate. PACS numbers: 03.65.Bz, 85.30.St. – 1 – Interest in quantum computing [1-27] has boosted studies of quantum mechanics of two-state systems such as the spin states of spin- 1 2 particles. We will use “spin ” to indicate a two-state system in this context. The “binary ” up and down spin states are of particular significance and

Related papers

Quantum signal splitting as entanglement due to three-spin interactions

Dmitry Mozyrsky, Vladimir Privman A

The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t + \\Delta t. For possible applications in quantum computing, we derive an explicit Hamiltonian to accomplish this process; it turns out to involve only three-spin x,y-component interactions.

•Sep 16, 1996•1 citation

Quantum Signal Splitting that Avoids Initialization of the Targets

Dima Mozyrsky, Vladimir Privman A

The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system (qubit, spin) into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t+Δt. In essence, the restriction on the initial target states is exchanged for uncertainty as to which two of the three qubits retain copies of the initial source state.

arXiv:quant-ph/9609010•Modern Physics Letters B•Dec 20, 1997•1 citation

Design of Gates for Quantum Computation: The Three-Spin XOR Gate in Terms of Two-Spin Interactions

Dima Mozyrsky, Vladimir Privman A, Steven P. Hotaling

We propose to design multispin quantum gates in which the input and output two-state systems (spins) are not necessarily identical. We describe the motivations for such studies and then derive an explicit general two-spin interaction Hamiltonian which accomplishes the quantum XOR gate function for a system of three spins: two input and one output.

arXiv:quant-ph/9610008•International Journal of Modern Physics B•Feb 20, 1998•6 citations

Universality in quantum computation

Jennifer L. Dodd

A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers.        The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected.       This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate.       Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.

•Jan 1, 2004•7 citations

State-Based Quantum Simulation: Releasing the Powers of Quantum States and Copies

S. Alipour, A. T. Rezakhani, Alireza Tavanfar +2 more

Quantum computing employs controllable interactions to perform sequences of logical gates and entire algorithms on quantum registers. This paradigm has been widely explored, e.g., for simulating dynamics of manybody systems by decomposing their Hamiltonian evolution in a series of quantum gates. Here, we introduce a method for quantum simulation in which the Hamiltonian is decomposed in terms of states and the resulting evolution is realized by only controlled-swap gates and measurements applied on a set of auxiliary systems whose quantum states define the system dynamics. These auxiliary systems can be identically prepared in an arbitrary number of copies of known states at any intermediate time. This parametrization of the quantum simulation goes beyond traditional gate-based methods and permits simulation of, e.g., state-dependent (nonlinear) Hamiltonians and open quantum systems. We show how classical nonlinear and time-delayed ordinary differential equations can be simulated with the state-based method, and how a nonlinear variant of shortcut to adiabaticity permits adiabatic quantum computation, preparation of eigenstates, and solution of optimization tasks.

arXiv:2505.13901•May 20, 2025•0 citations

Putting a Spin on Spin

Alice Flarend, Bob Hilborn

Abstract This chapter provides a brief introduction to the terminology and basic concepts of quantum spin systems, which are widely used in QIS and QC. The quantum spin of an elementary particle is an example of a qubit and can be visually represented as a spinning top. The vector direction of spin in state space, however, is different from the direction in physical space. Quantum spin-1/2 systems have two orthogonal states: spin-up and spin-down. The distinction between state space angles and physical space angles is emphasized. Because spin is a qubit, the quantum gates from Chapter 6 can be applied to spin-1/2 quantum states with predictable results.

•Apr 15, 2022•0 citations
We derive an explicit Hamiltonian for copying the basis up and down states of a quantum two-state system—a qubit—onto n “copy ” qubits (n ≥ 1) initially all prepared in the down state. In terms of spin components, for spin- 1 2 particle spin states, the resulting Hamiltonian involves n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a quantum-computing controlled-NOT gate. PACS numbers: 03.65.Bz, 85.30.St. – 1 – Interest in quantum computing [1-27] has boosted studies of quantum mechanics of two-state systems such as the spin states of spin- 1 2 particles. We will use “spin ” to indicate a two-state system in this context. The “binary ” up and down spin states are of particular significance and
The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t + \\Delta t. For possible applications in quantum computing, we derive an explicit Hamiltonian to accomplish this process; it turns out to involve only three-spin x,y-component interactions.
The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system (qubit, spin) into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t+Δt. In essence, the restriction on the initial target states is exchanged for uncertainty as to which two of the three qubits retain copies of the initial source state.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers.        The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected.       This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate.       Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
Quantum computing employs controllable interactions to perform sequences of logical gates and entire algorithms on quantum registers. This paradigm has been widely explored, e.g., for simulating dynamics of manybody systems by decomposing their Hamiltonian evolution in a series of quantum gates. Here, we introduce a method for quantum simulation in which the Hamiltonian is decomposed in terms of states and the resulting evolution is realized by only controlled-swap gates and measurements applied on a set of auxiliary systems whose quantum states define the system dynamics. These auxiliary systems can be identically prepared in an arbitrary number of copies of known states at any intermediate time. This parametrization of the quantum simulation goes beyond traditional gate-based methods and permits simulation of, e.g., state-dependent (nonlinear) Hamiltonians and open quantum systems. We show how classical nonlinear and time-delayed ordinary differential equations can be simulated with the state-based method, and how a nonlinear variant of shortcut to adiabaticity permits adiabatic quantum computation, preparation of eigenstates, and solution of optimization tasks.
Abstract This chapter provides a brief introduction to the terminology and basic concepts of quantum spin systems, which are widely used in QIS and QC. The quantum spin of an elementary particle is an example of a qubit and can be visually represented as a spinning top. The vector direction of spin in state space, however, is different from the direction in physical space. Quantum spin-1/2 systems have two orthogonal states: spin-up and spin-down. The distinction between state space angles and physical space angles is emphasized. Because spin is a qubit, the quantum gates from Chapter 6 can be applied to spin-1/2 quantum states with predictable results.