## Abstract

In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of unitary evolution, g , are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g = p ⊕ k . Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. The main contribution of the chapter is, we carry out a global analysis of time optimality.

### Keywords

- Kostant convexity
- spin dynamics
- Cartan decomposition
- Cartan subalgebra
- Weyl group
- time optimal control

## 1. Introduction

A rich class of model control problems arise when one considers dynamics of two coupled spin

A typical feature of many of these problems is that evolution of interaction Hamiltonians takes significantly longer than the time required to generate local unitary transformations (unitary transformations that effect individual spins only). In NMR spectroscopy [2, 3], local unitary transformations on spins are obtained by application of rf-pulses, whose strength may be orders of magnitude larger than the couplings between the spins. Given the Schróedinger equation for unitary evolution

where

The Hamiltonian of a spin

Note

where

The Hamiltonian for a system of two coupled spins takes the general form

where

is the coupling or interaction Hamiltonian and operates on both the spins.

The following notation is therefore common place in the NMR literature.

The operators

The unitary transformations of the kind

obtained by evolution of the local Hamiltonians are called local unitary transformations.

The coupling Hamiltonian can be written as

Written explicitly, some of these matrices take the form

and

The

for

The Lie algebra

Here

This decomposition of a real semi-simple Lie algebra

This special structure of Cartan decomposition arising in dynamics of two coupled spins in Eq. (1), motivates study of a broader class of time optimal control problems.

Consider the following canonical problems. Given the evolution

where

We assume

In general,

Given the Cartan decomposition

The special structure of this problem helps in complete description of the reachable set [27]. The elements of the reachable set at time

where

This reachable set can be understood as follows. The Cartan decomposition of the Lie algebra

Then, any arbitrary element of the group

for some

The results in this chapter suggest that

Written as evolution

where

The chapter is organized as follows. In Section 2, we study the *global analysis* of time optimality.

Given Lie algebra

## 2. Time optimal control for SU n / SO n problem

**Remark 1.** Birkhoff’s convexity states, a real

**Remark 2.** **KAK** decomposition in Eq. (17) states for

where

**Remark 3.** We now give a proof of the reachable set (16), for the

To understand the reachable set of this system we make a change of coordinates

If we understand reachable set of

**Theorem 1.** Let

and

**Proof.** As a first step, discretize the evolution of

For

By KAK,

where,

Observe

We equate

Multiplying both sides with

where,

We evaluate

such that

Given

with

With this choice of

Consider the case, when

where

Consider the decomposition

where

where

Then we write

where in Eq. (24) we can solve for

Let

where

is a diagonal matrix. Let

where

Note

as

Now using

where the above expression can be written as

where

Let

Choose an ordering of

By Schur convexity,

where

Therefore,

The difference

is regulated by size of

For each point

Then we get the following recursive relations.

where

Adding the above equations,

where

where

Note, **q.e.d.**

**Corollary 1.** Let

where

**Proof.** Let

From Theorem 1, we have

We can synthesize **q.e.d.**

**Remark 4.** We now show how Remark 2 and Theorem 1 can be mapped to results on decomposition and reachable set for coupled spins/qubits. Consider the transformation

The transformation maps the algebra

**Corollary 2. Canonical decomposition.** Given the decomposition of SU(4) from Remark 2, we can write

where

Multiplying both sides with

where

**Corollary 3. Digonalization.** Given

Note

where

**Corollary 4.** Given the evolution of coupled qubits

Furthermore

**Proof.** Let

Consider the product

where

Observe

Multiplying both sides with

which we can write as

where using

Furthermore **q.e.d.**

## 3. Time optimal control for G / K problem

**Remark 5. Stabilizer:** Let

Let

The range of

where

Let

then by one to one property of

Given a sequence

Let

if

as

**Remark 6. Kostant’s convexity:** [28] Given the decomposition

**Theorem 2** Given a compact Lie group

where

where

**Proof.** As in proof of Theorem 1, we define

and show that

where for

where

To show Eq. (46), we show there exists

where

Given

where

We bound the largest element (absolute value) of

where

Given decomposition of

Given

where

We describe an iterative procedure

where

where

Note

Where, using bound in

This gives,

For

Similarly,

Note,

where

The above exercise was illustrative. Now we use an iterative procedure as above to show Eq. (47).

Writing

where

We refer to Remark 5, Eq. (45). Given

which gives

we obtain

where

where

where, using bounds derived above

which gives

We can decompose,

This gives

For

Using

Similarly,

Note,

where

The above iterative procedure generates

where

such that

This follows because the orthogonal part of

(

**Lemma 1** Given

where

**Proof.** Note,

This implies

We have shown existence of

Applying the theorem again to

**Lemma 2** Given

where

where,

Using Lemma 1 and 2, we can express

Letting

Hence the proof of theorem. **q.e.d.**

## 4. Conclusion

In this chapter, we studied some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. We saw how dynamics was decomposed into fast generators