geometric algebra: imaginary numbers are real g eometric...

G e o m e t r i c A l g e b r a : I m a g i n a r y N u m b e r s A r e R e a l

Jan.

GEOMETRIC ALGEBRA:Imaginary Numbers Are Real

Timothy F. Havel (Nuclear Engineering)

LECTURE 2

On spin & spinors:

The operators for the , & components of the angularmomentum of a spin 1/2 particle (in units of ) are usuallyrepresented by one half the Pauli matrices:

These are easily seen to anticommute & square to the identity,and hence correspond to orthonormal vectors in a matrixrepresentation for .

The state of a spin is more commonly specified by a spinor(unit vector) in a 2-D Hilbert space over , in which casethe expected components of the angular momentum are

where , i.e. the inner product of !

x y zh

s10 11 0

∫ s20 i–

i 0∫ s3

1 00 1–

∫

G 3( )

y| Ò H ¬

12--- y· |sk y| Ò 1

2--- tr Ysk( ) 1

4--- tr Ysk skY+( ) 1

2--- tr Y sk∑( )1( ) Y sk∑= = = =

Y y| Ò y· |∫ Y sk, G 3( )�

T I M O T H Y F . H A V E L

L E C T U R E # 2

1 of 2828, 2002


Jan.

Since is a scalar, itself must be the sum of a scalar& a vector, and the scalar part is .Using the fact that the angular momentum is quantized along itsown axis to , the vector part can be written as ,i.e. as a rotation of by some spinor . It follows that

or, using the correspondence between and matrices:

Thus we can identify with , where isan idempotent ( ). In fact, since the 2nd column of

matrices is determined by the first, we can just use !

Another big advantage of representing states by sums ofscalars & vectors is that this representation extends directly tostatistical ensembles of spin states, as follows:

We may write this (dropping the bar) as , where is the ensemble’s polarization. The “operator” ,

called the density operator, yields the ensemble averageexpectation value of any observable via:

Y sk∑ YY· Ò 0

12--- tr Y( )∫ 1

2--- y y· | Ò 1

2---= =

1 2§ 12--- Rs3R

s3 R G+ 3( )�Y R 1

2--- 1 s3+( )( )R=

G+ 3( ) SU 2( )

Y a b*–

b a*1 00 0

a* b*b– a

a

ba* b* a 2 ab*

ba* b 2y| Ò y· |= = = =

y| Ò y RE+∫ E+12--- 1 s3+( )∫

E+2 E+=

SU 2( ) R

Y{ }

Y 1M----- Yii 1=

MÂ 12---

12M--------- Ri s3Rii 1=

MÂ+ 12--- R s3 R P R( )dÚ∞+æÆ= =

M �Æ

Y 1 s+( ) 2§=s s 1£∫ Y

A

y· | A y| Ò M 1– y· | A y| Òi 1=MÂ 2M 1– A Yi· Ò 0i 1=

MÂ 2 AY· Ò 0= = =


L E C T U R E # 2

2 of 2828, 2002


Jan.

Spins from Space-Time (sic)

The spin vector is the net angular momentum (in units of ),which for a classical spinning particle of charge leads to amagnetic moment of ( ), where is theBohr magneton. The spatial vector corresponds to a space-time bivector , where defines its inertial frame. In aco-moving frame, this evolves according to the covariant form ofLorentz equation (Lecture 3) as

,

where we have used the facts that anticommutes with , , and , while is a scalar but is a spatial vector.

The lack of response to the electric field is a result of choosing aco-moving frame. Thus the magnetic moment of the electron,which is very nearly , can be explained by a simple classicalmodel. Like the magnetic field itself, however, spin behavesunder inversion like a spatial bivector .

s he

sm0 se 2m( )§∫ c 1= m0s

s sg 0= g 0

s =

2m0 F s g 0( )∑( ) g 0� 2m0 E s∑ iB( ) s∑+( )g 0( ) g 0� 2m0s B¥= =

g 0 s EB i E s∑ iB( ) s∑

2m0

is


L E C T U R E # 2

3 of 2828, 2002


Jan.

MULTIPARTICLE ALGEBRA

More than the Sum of Its Parts?

To model a system of particles:

" Take as usual the direct sum of their states (vectors);

" Consider the corresponding joint Clifford algebra, i.e.

;

" This algebra has dimension (exponential in !).

Assuming there exists a common frame of reference, i.e. anatural choice of time-like in every particle space, then:

" The even subalgebras of different particle

spaces commute, since for and

where the superscripts are particle indices.

" Thus one has an isomorphism with the tensor productof the corresponding 3-D Euclidean geometric algebras:

;

" This algebra has dimension (still exponential!).

N

G 1 3,( )( )N G N 3N,( )ª

24N N

g 0m

G+ 1 3,( )

1 i j, 3£ £ 1 k l< N£ £

skmsl

n g kmg 0

m( ) g lng 0

n( )´ g lng 0

n( ) g kmg 0

m( )= slnsk

m´

G+ 1 3,( )( )N G 3( )( )�Nª

23N


L E C T U R E # 2

4 of 2828, 2002


Jan.

PRODUCT OPERATORSThe usual nonrelativistic theory of spin 1/2 particles reliesentirely upon the algebra of matrices over , which hasdimension only . The extra degrees of freedom in thealgebra are due to the fact that it contains a differentimaginary unit for every particle .

These extra degrees of freedom can be removed can beremoved by multiplication by the correlator idempotent:

This projects everything onto an “ideal” of the correct dimen-sion, in which all these different imaginary units have beenidentified. In addition, since is easily seen to commutewith everything in the algebra, multiplication by it is trivially ahomomorphism onto this ideal (subalgebra).

Henceforth we shall drop the implicit factor of from all ourexpressions, and use the single imaginary unit

.

Note the correlated product of the even subalgebras in eachparticle space, , is a subalgebra of dimension (the same as that of the subspace of Hermitian matrices), whichhas never been recognized in the matrix algebra!

N2N 2N¥ C

22N 1+

G+ 1 3,( )( )N

imm

C 12--- 1 i1i2

–( )12--- 1 i1i3

–( ) ºº 12--- 1 i1iN

–( )∫

C C2=

C

i i1C∫ º iNC= =

G + 3( )( )�N C§ 22N


L E C T U R E # 2

5 of 2828, 2002


Jan.

MULTIPARTICLE SPINORS

Just as in the one-particle case, multi-particle spinors (statevectors) in the -D complex Hilbert space correspond toelements of a left-ideal obtained by multiplying through by anidempotent:

.

This ideal corresponds, as before, to matrices whose columnsare all zero save for their left-most, and provides a carrier spacefor the products of all possible 3-D rotations.

Similarly, multiparticle density operators may now be written(omitting now the ) as

,

where . Note however that may not befactorizable into one particle rotations , in which casethe corresponding spinor cannot be expressed as aproduct of one-particle spinors and so is termedentangled. It may also happen that the individual terms in thissum can be factorized as , but the overall sum is notfactorizable. In this case the density operator describes acorrelated, but not an entangled, statistical ensemble.

y| Ò 2N

EC E+1ºE+

NC∫ E±

m 12--- 1 s3

m±( )∫( )

C

Y 1M----- Yii 1=

MÂ 1M----- yiyii 1=

MÂ 1M----- Ri E Rii 1=

MÂ= = =

Ri G+ 3( )( )�N C§� RiRi

1ºRiN

yi RiE=yi

1ºyiN

RimE+

m Rim


L E C T U R E # 2

6 of 2828, 2002


Jan.

THE BASICS OF QUANTUM INFORMATION PROCESSING

Binary information can be stored in an array of two-statequantum systems (e.g. spin 1/2) relative to a fixedcomputational basis of (or ideal in ):

Each such qubit can be placed into a coherent superpositionover these two basis states, which for can be written inthe following equivalent ways:

H 2N( ) G 3( )( )�N C§

0| Ò 10

= 0| Ò 0· | 1 00 0

= E+12--- 1 s3+( )=´ ´

1| Ò 01

= 1| Ò 1· | 0 00 1

= E-12--- 1 s3–( )=´ ´

a b, C�

a 0| Ò b 1| Ò+ab

= ´

a 0| Ò b 1| Ò+( ) a 0· | b 1· |+( ) a 2 abab b 2

= ´

a bs1+( )E+ a bs1+( ) 12--- 1 ¬ ab( )s1 ¡ ab( )s2+ +(=

a 2 b 2–( )s3+ ¯�


L E C T U R E # 2

7 of 2828, 2002


Jan.

A superposition over all qubits individually expands to asuperposition over all integers in , e.g.

,

( is obtained by binary expansion of the integer ).

By the linearity of quantum mechanics, any unitary operation on such a superposition operates on every term of its

expansion in parallel, i.e.

(in , acts as , which is also linear).Moreover, there exists a polynomially-bounded basis for theunitary group (i.e. one that is unitarily universal).

Example: All one-qubit operations, including those mappingone qubit states to superpositions, e.g. the Hadamard gate

,

together with the c-NOT (controlled-NOT)logic gate which computes the XOR of itsinputs (see right). Note that this operates onbit b conditional on bit a.

0 º 2N 1–, ,{ }

W| Ò 2N 2§ 0| Ò 1| Ò+( )ºº 0| Ò 1| Ò+( ) 2N 2§j| Ò

j 0=2N 1–Â= =

N Ï Ô Ô Ô Ô Ì Ô Ô Ô Ô Ó

j| Ò 01º0| Ò∫ j

U

U W| Ò 2N 2§U j| Ò

j 0=2N 1–Â=

G+ 3( )( )�N C§ U U W| Ò W· |U

U 2N( )

RH 0| Ò 0| Ò 1| Ò+( ) 2§= RH 1| Ò 0| Ò 1| Ò–( ) 2§=

≈

a

b

a

a + b


L E C T U R E # 2

8 of 2828, 2002


Jan.

There exists a subgroup of which:

" Maps single states to single states.

" Admits a polynomially bounded basis for the symmetricgroup (i.e. that is computationally universal).

This is nontrivial, since unitary operations are necessarilyreversible (i.e. each output corresponds to a single input).

Example: The Toffoli gate has threeinput and three output bits. It copies thea & b bits, but takes the NOT of the c bitif the other two are both 1 (see right).Setting the c bit to 1 thus takes the NANDof the other two.

Some of the hardest and most interesting computationalproblems (e.g. NP-complete) are decision problems:

Given a boolean function andan integer , decide if the set has some global property (e.g. is nonempty, periodic,etc.).

One might think such problems could be efficiently solved on aquantum computer by applying a unitary transformation (which computes the function ) to a superposition,

U 2N( )

j| Ò

S 2N( )

≈

a

c

a

(a * b) + c

b b

f : 0 1,{ }N 0 1,{ }MæÆ0 j 2M 1–£ £ i f i( ) j={ }

Uf

f


L E C T U R E # 2

9 of 2828, 2002


Jan.

,

followed by measurement of the output qubit’s state. Butaccording to von Neumann’s measurement postulate:

The result of a measurement on a quantum system in

a superposition is a random term in the

superposition, each with probability equal the

modulus of its coefficient.

For , this implies the chance of observing a given is

,

which may be exponentially small. Even if it is not small, some

properties of (e.g. periodicity) can only be found by

enumeration all or most of it.

There is a way around this: If one adjusts the phases of the

terms of the input superposition appropriately, an additional

transformation can be found that will amplify the magnitude of

the coefficients asso-ciated with the solution(s), such that the

probability of finding the sol-ution upon making a measurement

becomes appreciable. Because of an analogy with light

diffraction, this is usually called interference.

Uf W| Ò 0| Ò( ) 2N 2§i| Ò f i( )| Ò

i 0=2N 1–Â=

W| Ò 0| Ò j| Ò

Pr i f i( ) j=( ) i f i( ) j={ } 2N§=

i f i( ) j={ }


L E C T U R E # 2

10 of 2828, 2002


Jan.

)

Example: The Deutsch-Jozsa algorithm.

Suppose we are given a function , and ask:

" Is a constant function, i.e.

, or:

" Is a balanced function, i.e.

??

(two of the four possible functions areconstant, while the other two are balanced). This can bedetermined classically only by evaluating on both inputs.

1) Starting with , apply the Hadamard gates:

2) Evaluate via its unitary transformation :

where is the complement (or negation) of .

3) Using the fact that

f : 0 1,{ } 0 1,{ }Æ

f

i f i( ) 0={ } 0=( ) or i f i( ) 1={ } 0=( )

f

i f i( ) 0={ } i f i( ) 1={ }=

f : 0 1,{ } 0 1,{ }Æ

f

01| Ò 0| Ò 1| Ò∫

y1| Ò RH1

RH2( ) 01| Ò RH 0| Ò( ) RH 1| Ò( )∫ ∫ 1

2--- 0| Ò 1| Ò+( ) 0| Ò 1| Ò–( )=

f Uf

y2| Ò Uf y1| Ò∫ 12--- Uf 0| Ò 0| Ò Uf 1| Ò 0| Ò Uf 0| Ò 1| Ò Uf 1| Ò 1| Ò––+( )=

12--- 0| Ò f 0( )| Ò 1| Ò f 1( )| Ò 0| Ò f 0( )| Ò 1| Ò f 1( )| Ò––+(=

f 1 f–= f

f i( )| Ò f i( )| Ò–0| Ò 1| Ò– if f i( ) 0=

1| Ò 0| Ò– if f i( ) 1=Ó þÌ ýÏ ¸

1–( ) f i( ) 0| Ò 1| Ò–( )= =


L E C T U R E # 2

11 of 2828, 2002


Jan.

we can rewrite this as follows:

4) Now apply the same pair of Hadamard gates again:

5) This in turn may be rewritten as

showing the concentration of coherence via interference.

6) Thus measurement of the “input” (left-most) qubit gives if is constant and if is balanced.

The remarkable thing is that this took only a single evalua-tion ofthe function . This algorithm was the first hint that quantumcomputers are more powerful than classical.

NB: A general quantum search algorithm is known which isquadratically faster than classical linear search, but very fewquantum algorithms are known that are exponentially faster.

y2| Ò 12--- 1–( ) f 0( ) 0| Ò 1–( ) f 1( ) 1| Ò+( ) 0| Ò 1| Ò–( )=

y3| Ò RH1

RH2( ) y2| Ò∫ since RH 0| Ò 1| Ò–( ) 2 1| Ò=( )

12--- 1–( ) f 0( ) 0| Ò 1| Ò+( ) 1–( ) f 1( ) 0| Ò 1| Ò–( )+( ) 1| Ò=

y3| Ò1–( ) f 0( ) 0| Ò 1| Ò if f 0( ) f 1( )=

1–( ) f 0( ) 1| Ò 1| Ò if f 0( ) f 1( )=ÓÌÏ

=

0| Òf 1| Ò f

f


L E C T U R E # 2

12 of 2828, 2002


Jan.

The Geometry Behind the Logic

The Hadamard Gate RevisitedRecall the action of the Hadamard gate on basis states:

,

The idempotents corresponding to the resulting states are:

We can also show that

which becomes on multiplying thru by the phasefactor . The corresponding idempotents are:

Since and maps & , it follows that is just a rotation by about the axis !

RH 0| Ò 0| Ò 1| Ò+( ) 2§= RH 1| Ò 0| Ò 1| Ò–( ) 2§=

RHE+ RH12--- 1 1

1 1 = G+ 1 s1+( ) 2§∫´

RHE- RH12--- 1 1 –

1– 1 = G- 1 s1–( ) 2§∫´

RH 0| Ò i 1| Ò+( ) 2§ 0| Ò 1| Ò+( ) i 0| Ò 1| Ò–( )( )+( ) 2§=

1 i+( ) 0| Ò 1 i–( ) 1| Ò+( ) 2§=

0| Ò i 1| Ò–( ) 2§1 i–( ) 2§

0| Ò i 1| Ò±( ) 0· | i 1· |±( ) 12--- 1 i+-

i± 1 = F± 1 s2±( ) 2§∫´

RH( )2 1= x z´ y y–´ RHp h s1 s3+( ) 2§=


L E C T U R E # 2

13 of 2828, 2002


Jan.

Recall such rotations can be written in exponential form as

(since ).

It may be observed that , due to a geometricallyirrelevant global phase (a gauge degree of freedom); the phasecorrected Hadamard gate is

; similarly,

the NOT operation is a rotation by about , i.e. .

The c-NOT Gate RevisitedRecall that the c-NOT gate takes the NOT of one qubit if theother is , e.g.

Using the nilpotent transition operators & , we canwrite this as

i p 2§( )h–( )exp ih–= h2 1=

ih–( )2 1–=

RH i p 2§( ) 1 h–( )( )exp h= =

N p s1 N s1=

1| Ò

S2 1

00| Ò 00| Ò= S2 1

01| Ò 01| Ò=

S2 1

10| Ò 11| Ò= S2 1

11| Ò 10| Ò=

0| Ò 1· | 1| Ò 0· |

S2 1

00| Ò 00· | 01| Ò 01· | 11| Ò 10· | 10| Ò 11· |+ + +=

00| Ò 00· | 01| Ò 01· | s12 10| Ò 10· | 11| Ò 11· |+( )+ +=

0| Ò 0· |( ) 0| Ò 0· | 1| Ò 1· |+( ) s12 1| Ò 1· |( ) 0| Ò 0· | 1| Ò 1· |+( )+=

0| Ò 0· | 1� 1| Ò 1· | 1�( )s12

+ E+1

E-1s1

2+´=


L E C T U R E # 2

14 of 2828, 2002


Jan.

This has an interesting ideal-theoretic interpretation:

Rotate qubit 2 by in the left-ideal defined by , but

leave it alone in the complementary ideal defined by

the annihilating idempotent .

Using a well-known formula for the exponential of any-thingtimes a commuting idempotent, this may be written as

More generally, a conditional rotation is a rotation of a qubit inthe left-ideal generated by the idempotents of the given states ofone or more other qubits.

Noting that so that rotations about in the

ideal generated by are also conditional phase shifts, the form

makes it clear that can alternatively be viewed

as a phase shift of the first qubit conditional on the second being

along . We can therefore write as a phase shift

conditional on both qubits being along , sandwiched between

two Hadamards, as follows:

p E-1

E+1

i p 2§( )E-1 1 s1

2–( )( )exp 1 E-

1–( ) E-

1 i p 2§( ) 1 s12

–( )–( )exp+=

E+1

E-1s1

2+ 1 2E-

1G-

2 .–= =

is3E± iE±±= s31

G-2

1 2E-1G-

2– S2 1

x– S2 1

z–

RH2 ipE-

1E-

2–( )exp RH

2 ipE-1

RH2

E-2

RH2

–( )exp ipE-1G-

2–( )exp= =


L E C T U R E # 2

15 of 2828, 2002


Jan. 2

Using green & red arrows for the qubit projected onto the & ideals, this can be represented by vector diagrams asfollows:

This is actually rather close to how is implemented in NMRspectroscopy, which also makes common use of such dia-grams without recognizing their ideal-theoretic connection.

The Deutsch-Jozsa Algorithm Revisited

The initial superposition in the Deutsch-Jozsa algorithm is

If , and so the final Hadamards just

transform this back to , telling us it’s constant.

E-1

E+1

x

y

z

RH

RHP.S.

S2 1

y1| Ò y1· | G+1G-

2 14--- 1 s1

1 s12

– s11s1

2–+( )= =

f 0( ) f 1( ) 0= = Uf 1=

E+1E-

2


L E C T U R E # 2

16 of 288, 2002


Jan.

Now suppose we have the function , .Then , i.e. the output qubit is flipped from to when the input bit is . How does this act on the productoperators in the initial state? To answer that question bygeometric reasoning only, let us go into the ideals generated by

and , where we find that

.

This shows that this state is that shown on theright, i.e. the two components of the first qubitpoint in opposite directions along . We knowthat will rotate the greenone by about to give us the state:

.

Thus the net effect of is to do a swap,

;

applied to , it swaps the signs of these terms (while it

commutes with & hence has no effect on ), so we get:

The final Hadamards convert this to as expected.

f 0( ) 0= f 1( ) 1=

Uf S2 1= 0| Ò 1| Ò1| Ò

G-2 G+

2

s11s1

2 s11s1

2G+

2G-

2+( ) s1

1G+

2 s11G-

2–= =

x

S2 1 1 2E-1G-

2–=

p z

s11G+

2 s11G-

2+ s1

1=

S2 1

s11 s1

1s12´

y1| Ò y1· |s1

2

S2 1 y1| Ò y1· |S2 1 14--- 1 s1

1– s1

2– s1

1s12

+( ) G-1G-

2= =

E-1

E-2


L E C T U R E # 2

17 of 2828, 2002


Jan.

CLOSING REMARKSThe Future of Quantum Computing

1) To date, computers have done more than theory has inenabling us to deal with complexity in the natural sciences(and elsewhere), because very simple local dynamics canmodel arbitrarily complex global behavior.

2) The situation in multiparticle quantum systems is similar:Simple (indeed linear) local dynamics can lead to verycomplex large-scale behavior, including the apparent non-linearities of the classical world.

3) Quantum computers may yield only modest advantages incombinatorial problems, but they will certainly provide apowerful means of simulating general quantum systems.

4) Thus the advent of quantum computers will greatly extendthe reach of fundamental physics to the complex systemsfound in chemistry and molecular biology.

5) Meanwhile, there is much to be learned about how simplequantum dynamics leads to classical complexity (evenchaos), which is the subject of the theory of decoherence.

6) By computing with homomorphic images in Clifford alge-bras, this may even enable theorems to be “proved” by exp-eriment (for CaF2, a dimension of has been reached).21011


L E C T U R E # 2

18 of 2828, 2002


Jan.

THE BASICS OF NMRA macroscopic quantum system:Although the magnetic dipoles of near-by spins interact, the rapidrotational diffusion of the molecules in liquids average theseinteractions to zero. Hence to an excellent first-order approx-imation, spins in different molecules do not interact. It follows thatif the state of the spins in the -th molecule is , then densityoperator of an molecule ensemble factorizes as follows:

Thus the kinematics is identical to that of a single molecule!

The dynamics and observables are also identical, since:

m ym| ÒM

Y y1y2ºyM| Ò y1y2ºyM· |=

y1| Ò y1· | º yM| Ò y M· | y1| Ò y1· | º yM| Ò y M· |= =

Y1Y2ºYM Y�M Y M1– ym| Ò ym· |

mÂ∫( )ª=

)

e i– tSmHm

Y�M( ) eitSmHm

e i– tHYeitH( )�M

=

tr Y�M sam

mÂË ¯Ê � M tr Ysa

m( )=


L E C T U R E # 2

19 of 2828, 2002


Jan.

The weak-coupling Hamiltonian:

The Hamiltonian of liquid-state NMR has the form:

The first term is the Zeeman interaction (in rad/sec) with theexternal magnetic field (along ) as before; the second is thescalar coupling of pairs of spins across chemical bonds.

It follows from first-order perturbation theory that if (weak coupling), the scalar coupling

terms may be replaced by their secular parts . Nowsince is diagonal, may be given in closed form.

Radio-frequency fields:

Given a strong RF-field on-resonance with the -th spin,

with , it follows from the Liouville-von Neumannequation that in a co-rotating frame

,

the (spin vector of) the spin rotates about the axis accordingto . Henceforth, all our transform-ations will bereferred to such a rotating frame (w/o primes).

H 12--- w0

k s3k

kÂ– p2---- J

kl s1k s1

l s2k s2

l s3k s3

l+ +( )

k l<Â+=

z

w0k w0

l– >> pJkl k l,"pJkls3

k s3l

H itH–( )exp

k

HRF w1k w0

kt( )s1

kcos w0

kt( )sin s2

k+( ) w1

k iw0kts3

k( )exp s1k

= =

w1k >> p Jkl l"

Y¢ iw0kts3

k–( )exp Y iw0

kts3

k( )exp=

x¢iw1

kt s1¢

k–( )exp


L E C T U R E # 2

20 of 2828, 2002


Jan.

In-Phase & Anti-phase Coherence

The effect of scalar coupling:The (weak) scalar coupling propagator has the form:

Applied to a transverse state of e.g. spin , this yields:

This is a rotation between in-phase ( ) and anti-phase ( )coherence on spin .

A classical interpretation is found on rotating to the -axis,where the diagonal matrix elements in the basis ( ,

) are deviations from equal populations:

The anti-phase population difference between & states ofspin is inverted in the subensemble where spin is down.

ipJ12

ts31s3

2 2§–( )exp pJ12

t 2§( )cos i pJ12

t 2§( )s31s3

2sin–=

1

i2pJ12

ts31s3

2 2§–( )s11 i2pJ

12ts3

1s32 2§( )expexp

pJ12

t( )s11

cos pJ12

t( )s21s3

2sin+=

s11 s2

1s32

1

z

s3 0| Ò ≠| Ò∫1| Ò Ø| Ò∫

PO 00· |PO 00| Ò 10· |PO 10| Ò 01· |PO 01| Ò 11· |PO 11| Ò

s31 1 1– 1 1–

s31s3

2 1 1– 1– 1

0| Ò 1| Ò1 2


L E C T U R E # 2

21 of 2828, 2002


Jan.

Spectra & vector diagrams:

Thus in- / anti-phase coherence corresponds to classicalensembles, characterized by these relative populations, whichare in transition between spin “up” & “down”. Moreover, spin

’s magnetization (population difference) precesses at the rate as spin is “up” or “down”, resp. The spectrum

(real part of the Fourier transform) is,

in which the anti-phase population inversion may be seen.

The rotation of into may be visualized as follows:

11w0

1 2p( )§ J12± 2

s11 s2

1s32

s11 s2

1s32

s11 s1

1–

s11s3

2 Magnetization of spin 1 in subensemble with spin 2 “up” & “down”.x

z

y


L E C T U R E # 2

22 of 2828, 2002


Jan.

The one-bit quantum logic gates:

The simplest logic gate is the NOT, which is a -rotation aboutthe x-axis combined with a phase shift,

;

recalling that the density operators of the basis states of the -th spin (with a totally mixed state everywhere else) are

,

we can use the anticommutivity of & to show that this NOTgate maps the state of the -th spin to the state:

# Another important one-bit, but nonboolean, gate is theHadamard transform HAD:

,

which can be show to map & . It also maps

& , and so can be usedto prepare a uniform superposition over all spins as above.

p

i p2--- 1 s1

k–( )Ë ¯

Ê �exp i p 2§( )cos i p 2§( )s1k

sin–( ) s1k

= =

k

1 º 0| Ò 0· | º 1� � � �( ) 12--- 1 s3

k+( ) E+

k∫=

1 º 1| Ò 1· | º 1� � � �( ) 12--- 1 s3

k–( ) E-

k∫=

s1k s3

k

0| Ò k 1| Ò

s1k E+

k s1k s1

k 12--- 1 s3

k+( )s1

k s1k( )

212--- 1 s3

k–( ) E-

k= = =

i p2--- 1 2 s1

k s3k

+( )–( )Ë ¯Ê �exp 2 s1

k s3k

+( )=

s1k s3

k´ s2k s2

k–´

0| Ò 0| Ò 1| Ò+( ) 2§´ 1| Ò 0| Ò 1| Ò–( ) 2§´


L E C T U R E # 2

23 of 2828, 2002


Jan.

The multi-bit boolean logic gates:Interesting computations require feedback, i.e. the state of onebit must influence what happens to another, but the usual AND& OR gates are not reversible (only one output!).

An important two-bit gate is the controlled-NOT:

This “c-NOT” is readily shown to flip the first spin in states wherethe second is down (just as we considered earlier), i.e.

,

or more compactly: . Thus idem-potents also describe the conditionality of operations.

Another obvious gate is the SWAP of two bits,

;

is also called the particle interchange operator.

These can be extended to bits; e.g., the Toffoli gate is:

The TOF alone is universal for boolean logic; more generally, thec-NOT and one-bit rotations generate all of .

i p2--- 1 s1

1–( )E-

2Ë ¯Ê �exp s1

1E-2 E+

2+=

00| Ò 00| Ò´ 10| Ò 10| Ò´ 01| Ò 11| Ò´ 11| Ò 01| Ò´

d1d2| Ò d1 d2 mod 2+( ) d2( )| Ò´

i p2---P12

Ë ¯Ê �exp P12 1

2--- 1 s1

1 s12 s2

1 s22 s3

1 s32

+ + +( )∫=

P12

N

i p2--- 1 s1

1–( )E-

2E-3

Ë ¯Ê �exp s1

1E-

2E-3 1 E-

2E-3

–( )+=

SU 2N( )


L E C T U R E # 2

24 of 2828, 2002


Jan. 2

NMR Implementations of Gates

Radio-frequency pulse sequences:The NOT gate is easily implemented by a strong RF pulse, inphase with the x-axis, whose frequency range spans only theresonance of the target spin, and whose duration is sufficient torotate it by (note the global phase offset of has no effect onthe density operator).

The HAD gate is similarly obtained from the pulsesequence (written in left-to-right temporal order):

To implement the c-NOT gate, we proceed as follows:

This is a -rotation about y, a weak coupling evolution for

, a -rotation about x, and a Zeeman evolution.

p i

p8---s2

k p2---s1

k p8---s2

k–Æ Æ i p

8---s2

k

Ë ¯Ê �exp i–

p2---s1

k

Ë ¯Ê �exp i–

p8---s2

k

Ë ¯Ê �exp¤

i p2--- 1 s1

1–( )E-

2Ë ¯Ê �exp i–

p4---s2

1Ë ¯Ê �exp ipE-

1E-2( )exp i p

4---s2

1Ë ¯Ê �exp=

i–p4---s2

1Ë ¯Ê �exp i–

p4--- s3

1 s32

+( )Ë ¯Ê �exp i p

4---s3

1s32

Ë ¯Ê �exp i p

4---s2

1Ë ¯Ê � iexp=

i–p4--- s3

1 s32

+( )Ë ¯Ê �exp i–

p4---s1

1Ë ¯Ê �exp i p

4---s3

1s32

Ë ¯Ê �exp i p

4---s2

1Ë ¯Ê � iexp=

p 2§( )1 2J12( )§ p 2§( )


L E C T U R E # 2

25 of 288, 2002


Jan.

Vector diagram description:# This pulse sequence may be depicted as follows:

In terms of Bloch diagrams, we have:

Caption: Here, red is magnetization of 1 spin in moleculeswhere 2 spin is up, and green that of 1 spin where 2 down.

1

2

1/(4J) 1/(4J) 1/(4n1) 1/(4n2)

y x x xx

$ % & '

x

y

z

$

%

&

'


L E C T U R E # 2

26 of 2828, 2002


Jan.

PSEUDO-PURE STATES

Starting from equilibrium:

Since , the equilibrium state of ahomonuclear spin system (& its partition function ) is:

Here is a binary expansion of , & where is the Hadamard weight (number of 1’s) of .

The problem with is that logical operations performed onthe spins at the microscopic level do not effect the sameoperations on their macroscopic polarizations; for two spins:

Spin 1’s polarization, i.e. the alternating row sum, goes to 0.

2pJ << w0 << kBTQ 2Nª

Yeq

HZ kBT§–( )exp

tr HZ kBT§–( )exp( )------------------------------------------------

w0l s3

llÂ– 2kBT§( )exp

Q----------------------------------------------------------= =

1 w0l s3

llÂ 2kBT§–( ) 2 N– 1

w0 pi i| Ò i· |iÂ

kBT-------------------------------–

Ë ¯Á �Ê �

2 N–ª ª

0101º0| Ò i| Ò pi h i( ) N 2§–=h i( ) i

Yeq

req

req¢

PO 00· |PO 00| Ò 10· |PO 10| Ò 01· |PO 01| Ò 11· |PO 11| Ò

s31 s3

2+ 1 0 0 1–

E+1s3

2 1 0 1– 0


L E C T U R E # 2

27 of 2828, 2002


Jan.

So we average yet more!

A pseudo-pure state is one whose density operator has exactlyone nondegenerate eigenvalue, e.g.

.

Note that the microscopic state is canonicallyassociated with .

Because the identity component is unitarily invariant, thestate provides a spinorial representation of :

Similarly, because the identity component does not contribute tothe magnetization (population differences), the ensembleaverage expectation value of the observables is proportional totheir ordinary expectation values:

Since their eigenstructures differ, must be prepared from by a nonunitary process, e.g. by averaging the popu-lations

over all permutations of the states ( ) (more efficientmethods exist, which rely upon magnetic gradients).

Ypp 1w0 p0 0| Ò 0· |

kBT---------------------------–Ë ¯

Ê � 2 N– 1 a 0| Ò 0· |+( ) 2 N–∫=

0| Ò 00º0| Ò=rpp

i| Ò SU 2N( )

UYppU 1 a U 0| Ò( ) 0· |U( )+( ) 2 N–= U SU 2N( )�( )

12---tr Ypps1

k( ) tr s1k( ) a tr s1

k 0| Ò 0· |( )+( ) 2 N 1–– a2N 1+-------------- 0· |s1

k 0| Ò= =

YppYeq

i| Ò i· | i 0>


L E C T U R E # 2

28 of 2828, 2002

geometric algebra: imaginary numbers are real g eometric...

Documents