arXiv:0808.3212 · Physical Review A

Quantum control via geometry: An explicit example

Mile Gu, Andrew C. Doherty, Michael A. Nielsen

Published Sep 23, 2008• 18 citations

Abstract

We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of $\mathrm{su}({2}^{n})$ into orthogonal subspaces $\mathfrak{l}$ and $\mathfrak{p}$ such that $[\mathfrak{l},\mathfrak{l}]\ensuremath{\subseteq}\mathfrak{p},[\mathfrak{p},\mathfrak{l}]=\mathfrak{p},[\mathfrak{p},\mathfrak{p}]\ensuremath{\subseteq}\mathfrak{l}$. Motion in the $\mathfrak{l}$ direction is assumed to have negligible cost, where motion in the $\mathfrak{p}$ direction does not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.

DOI: 10.1103/physreva.78.032327

Related papers

Quantum geometry of the Cartan control problem

Peter Leifer

arXiv:0810.3188Oct 17, 20080 citations

Solving the $KP$ problem with the Global Cartan Decomposition

Elija Perrier, Christopher S. Jackson

arXiv:2404.02358Apr 2, 20240 citations

The optimal control problem on SO(4) and its applications to quantum control

Domenico D’Alessandro

IEEE Transactions on Automatic ControlJan 1, 200237 citations

Geometric methods in quantum computation

Jun Zhang, Shankar Sastry

Jan 1, 20031 citation

Optimal synthesis of general multi-qutrit quantum computation

Gui‐Long Jiang, Wen‐Qiang Liu, Hai‐Rui Wei

arXiv:2310.11996Oct 18, 20230 citations

Optimal control, geometry, and quantum computing

Michael A. Nielsen, Mark R. Dowling, Mile Gu +1 more

arXiv:quant-ph/0603160Physical Review AJun 16, 2006137 citations

arXiv:0808.3212 · Physical Review A

Quantum control via geometry: An explicit example

Mile Gu, Andrew C. Doherty, Michael A. Nielsen

Published Sep 23, 2008• 18 citations

arXiv HTML PDF

Abstract

We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of

\mathrm{su}({2}^{n})

into orthogonal subspaces

\mathfrak{l}

and

\mathfrak{p}

such that

[\mathfrak{l},\mathfrak{l}]\ensuremath{\subseteq}\mathfrak{p},[\mathfrak{p},\mathfrak{l}]=\mathfrak{p},[\mathfrak{p},\mathfrak{p}]\ensuremath{\subseteq}\mathfrak{l}

. Motion in the

\mathfrak{l}

direction is assumed to have negligible cost, where motion in the

\mathfrak{p}

direction does not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.

DOI: 10.1103/physreva.78.032327

Related papers

Quantum geometry of the Cartan control problem

Peter Leifer

The Cartan control problem of the quantum circuits discussed from the differential geometry point of view. Abstract unitary transformations of $SU(2^n)$ are realized physically in the projective Hilbert state space $CP(2^n-1)$ of the n-qubit system. Therefore the Cartan decomposition of the algebra $AlgSU(2^n-1)$ into orthogonal subspaces $h$ and $b$ such that $[h,h] \subseteq h, [b,b] \subseteq h, [b,h] \subseteq b$ is state-dependent and thus requires the representation in the local coordinates.The Cartan control problem of the quantum circuits discussed from the differential geometry point of view. Abstract unitary transformations of $SU(2^n)$ are realized physically in the projective Hilbert state space $CP(2^n-1)$ of the n-qubit system. Therefore the Cartan decomposition of the algebra $AlgSU(2^n-1)$ into orthogonal subspaces $h$ and $b$ such that $[h,h] \subseteq h, [b,b] \subseteq h, [b,h] \subseteq b$ is state-dependent and thus requires the representation in the local coordinates.

arXiv:0810.3188Oct 17, 20080 citations

Solving the $KP$ problem with the Global Cartan DecompositionSolving the $KP$ problem with the Global Cartan Decomposition

Elija Perrier, Christopher S. Jackson

Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called $KP$ -problem, where target unitaries belong to a semi-simple Lie group manifold $G$ whose Lie algebra admits a $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in $\mathfrak{p}$ . In this paper, we propose a new method utilising global Cartan decompositions $G=KAK$ of symmetric spaces $G/K$ for generating time-optimal unitaries for targets $-iX \in [\frak{p},\frak{p}] \subset \frak{k}$ with controls $-iH(t) \in \frak{p}$ . Target unitaries are parametrised as $U=kac$ where $k,c \in K$ and $a = e^{iΘ}$ with $Θ\in \frak{a}$ . We show that the assumption of $dΘ=0$ equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by $\mathfrak{p}$ and local rotations are generated by $[\mathfrak{p},\mathfrak{p}]$ .Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called $KP$-problem, where target unitaries belong to a semi-simple Lie group manifold $G$ whose Lie algebra admits a $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in $\mathfrak{p}$. In this paper, we propose a new method utilising global Cartan decompositions $G=KAK$ of symmetric spaces $G/K$ for generating time-optimal unitaries for targets $-iX \in [\frak{p},\frak{p}] \subset \frak{k}$ with controls $-iH(t) \in \frak{p}$. Target unitaries are parametrised as $U=kac$ where $k,c \in K$ and $a = e^{iΘ}$ with $Θ\in \frak{a}$. We show that the assumption of $dΘ=0$ equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by $\mathfrak{p}$ and local rotations are generated by $[\mathfrak{p},\mathfrak{p}]$.

arXiv:2404.02358Apr 2, 20240 citations

The optimal control problem on SO(4) and its applications to quantum control

Domenico D’Alessandro

We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4/spl times/4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4/spl times/4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.

IEEE Transactions on Automatic ControlJan 1, 200237 citations

Geometric methods in quantum computation

Jun Zhang, Shankar Sastry

Recent advances in the physical sciences and engineering have created great hopes for new computational paradigms and substrates. One such new approach is the quantum computer, which holds the promise of enhanced computational power. Analogous to the way a classical computer is built from electrical circuits containing wires and logic gates, a quantum computer is built from quantum circuits containing quantum wires and elementary quantum gates to transport and manipulate quantum information. Therefore, design of quantum gates and quantum circuits is a prerequisite for any real application of quantum computation. In this dissertation we apply geometric control methods from differential geometry and Lie group representation theory to analyze the properties of quantum gates and to design optimal quantum circuits. Using the Cartan decomposition and the Weyl group, we show that the geometric structure of nonlocal two-qubit gates is a 3-Torus. After further reducing the symmetry, the geometric representation of nonlocal gates is seen to be conveniently visualized as a tetrahedron. Each point in this tetrahedron except on the base corresponds to a different equivalent class of nonlocal gates. This geometric representation is one of the cornerstones for the discussion on quantum computation in this dissertation. We investigate the properties of those two-qubit operations that can generate maximal entanglement. It is an astonishing finding that if we randomly choose a two-qubit operation, the probability that we obtain a perfect entangler is exactly one half. We prove that given a two-body interaction Hamiltonian, it is always possible to explicitly construct a quantum circuit for exact simulation of any arbitrary nonlocal two-qubit gate by turning on the two-body interaction for at most three times, together with at most four local gates. We also provide an analytic approach to construct a universal quantum circuit from any entangling gate supplemented with local gates. Closed form solutions have been derived for each step in this explicit construction procedure. Moreover, the minimum upper bound is found to construct a universal quantum circuit from any Controlled-Unitary gate. A near optimal explicit construction of universal quantum circuits from a given Controlled-Unitary is provided. For the Controlled-NOT and Double-CNOT gate, we then develop simple analytic ways to construct universal quantum circuits with exactly three applications, which is the least possible for these gates. We further discover a new quantum gate (named B gate) that achieves the desired universality with minimal number of gates. Optimal implementation of single-qubit quantum gates is also investigated. Finally, as a real physical application, a constructive way to implement any arbitrary two-qubit operation on a spin electronics system is discussed.Recent advances in the physical sciences and engineering have created great hopes for new computational paradigms and substrates. One such new approach is the quantum computer, which holds the promise of enhanced computational power. Analogous to the way a classical computer is built from electrical circuits containing wires and logic gates, a quantum computer is built from quantum circuits containing quantum wires and elementary quantum gates to transport and manipulate quantum information. Therefore, design of quantum gates and quantum circuits is a prerequisite for any real application of quantum computation. In this dissertation we apply geometric control methods from differential geometry and Lie group representation theory to analyze the properties of quantum gates and to design optimal quantum circuits. Using the Cartan decomposition and the Weyl group, we show that the geometric structure of nonlocal two-qubit gates is a 3-Torus. After further reducing the symmetry, the geometric representation of nonlocal gates is seen to be conveniently visualized as a tetrahedron. Each point in this tetrahedron except on the base corresponds to a different equivalent class of nonlocal gates. This geometric representation is one of the cornerstones for the discussion on quantum computation in this dissertation. We investigate the properties of those two-qubit operations that can generate maximal entanglement. It is an astonishing finding that if we randomly choose a two-qubit operation, the probability that we obtain a perfect entangler is exactly one half. We prove that given a two-body interaction Hamiltonian, it is always possible to explicitly construct a quantum circuit for exact simulation of any arbitrary nonlocal two-qubit gate by turning on the two-body interaction for at most three times, together with at most four local gates. We also provide an analytic approach to construct a universal quantum circuit from any entangling gate supplemented with local gates. Closed form solutions have been derived for each step in this explicit construction procedure. Moreover, the minimum upper bound is found to construct a universal quantum circuit from any Controlled-Unitary gate. A near optimal explicit construction of universal quantum circuits from a given Controlled-Unitary is provided. For the Controlled-NOT and Double-CNOT gate, we then develop simple analytic ways to construct universal quantum circuits with exactly three applications, which is the least possible for these gates. We further discover a new quantum gate (named B gate) that achieves the desired universality with minimal number of gates. Optimal implementation of single-qubit quantum gates is also investigated. Finally, as a real physical application, a constructive way to implement any arbitrary two-qubit operation on a spin electronics system is discussed.

Jan 1, 20031 citation

Optimal synthesis of general multi-qutrit quantum computation

Gui‐Long Jiang, Wen‐Qiang Liu, Hai‐Rui Wei

Quantum circuits of a general quantum gate acting on multiple $d$ -level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group $U(3^n)$ (arbitrary $n$ -qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the $n$ -qutrit system, and the quantum circuit of generic $n$ -qutrit gates contained $\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})$ GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.Quantum circuits of a general quantum gate acting on multiple $d$-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group $U(3^n)$ (arbitrary $n$-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the $n$-qutrit system, and the quantum circuit of generic $n$-qutrit gates contained $\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})$ GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.

arXiv:2310.11996Oct 18, 20230 citations

Optimal control, geometry, and quantum computing

Michael A. Nielsen, Mark R. Dowling, Mile Gu +1 more

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, $U$ , to the optimal control cost associated to the synthesis of $U$ . These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, $U$, to the optimal control cost associated to the synthesis of $U$. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.

arXiv:quant-ph/0603160Physical Review AJun 16, 2006137 citations

We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of

\mathrm{su}({2}^{n})

into orthogonal subspaces

\mathfrak{l}

and

\mathfrak{p}

such that

[\mathfrak{l},\mathfrak{l}]\ensuremath{\subseteq}\mathfrak{p},[\mathfrak{p},\mathfrak{l}]=\mathfrak{p},[\mathfrak{p},\mathfrak{p}]\ensuremath{\subseteq}\mathfrak{l}

. Motion in the

\mathfrak{l}

direction is assumed to have negligible cost, where motion in the

\mathfrak{p}

direction does not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.

The Cartan control problem of the quantum circuits discussed from the differential geometry point of view. Abstract unitary transformations of

SU(2^n)

are realized physically in the projective Hilbert state space

CP(2^n-1)

of the n-qubit system. Therefore the Cartan decomposition of the algebra

AlgSU(2^n-1)

into orthogonal subspaces

h

and

b

such that

[h,h] \subseteq h, [b,b] \subseteq h, [b,h] \subseteq b

is state-dependent and thus requires the representation in the local coordinates.

Solving the

KP

problem with the Global Cartan Decomposition

KP

-problem, where target unitaries belong to a semi-simple Lie group manifold

G

whose Lie algebra admits a

\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}

decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in

\mathfrak{p}

. In this paper, we propose a new method utilising global Cartan decompositions

G=KAK

of symmetric spaces

G/K

for generating time-optimal unitaries for targets

-iX \in [\frak{p},\frak{p}] \subset \frak{k}

with controls

-iH(t) \in \frak{p}

. Target unitaries are parametrised as

U=kac

where

k,c \in K

and

a = e^{iΘ}

with

Θ\in \frak{a}

. We show that the assumption of

dΘ=0

equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by

\mathfrak{p}

and local rotations are generated by

[\mathfrak{p},\mathfrak{p}]

Quantum circuits of a general quantum gate acting on multiple

d

-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group

U(3^n)

(arbitrary

n

-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the

n

-qutrit system, and the quantum circuit of generic

n

-qutrit gates contained

\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})

GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation,

U

, to the optimal control cost associated to the synthesis of

U

. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.