CROSS-REFERENCE TO RELATED APPLICATION

This application is based on Provisional application No. 60/107,762 filed Nov. 10, 1998 which is herein incorporated by reference.

FIELD OF THE INVENTION

This invention pertains generally to quantum computers, and in particular to a solid state quantum computer comprising a crystal lattice.

BACKGROUND

A quantum computer is a machine that prepares a quantum mechanical system in an initial input state, performs unitary logic operations on the system, and measures a resulting output state. The superposition principle of quantum mechanics and the quantum interference induced by a projective measurement bring about “quantum parallelism,” by which certain problems can be computed faster than by any classical computer.

Several physical systems are known to provide the quantum mechanical states that carry quantum bit, or “qubit,” information. Each of these systems carries its own disadvantages. (i) Cold trapped ions, cavity quantum electrodynamics, and combinations of the two systems allow computation and final readout to be performed on individual qubits. However, these techniques require immense experimental exertion. (ii) Single photon gates, compatible with quantum communication links, are possible only in the presence of large optical nonlinearity with negligible loss. (iii) Systems of nuclear spins in solution molecules are based on natural and simple chemical structures and well-established techniques of pulsed nuclear magnetic resonance. However, such systems demand a large number of molecules to detect the output signals from slight population differences between nuclei with up spins and those with down spins. Furthermore, these liquid systems cannot be cooled to low temperatures to reduce thermal noise, since the liquids would freeze. (iv) Solid state systems can be used, including quantum dots, Josephson junctions, and nuclear spins of implanted phosphorous ions in silicon. However, these solid state systems would require technical breakthroughs before they could be scaled up to large numbers of distinct qubits.

There is a need, therefore, for a quantum computer that does not require cumbersome experimental techniques, that can be cooled to low temperatures, and that can be scaled to large numbers of qubits.

A solid state crystal can be considered for quantum computation, wherein qubits are represented by spins of individual atoms or nuclei. However, it is not obvious how to distinguish between adjacent spins in the crystal, since the spins are so close together (on the order of 10 angstroms). Furthermore, spurious couplings among the atoms of the crystal will contaminate the quantum computation.

OBJECTS AND ADVANTAGES

It is therefore a primary object of the present invention to provide a crystal lattice quantum computer wherein quantum bits are represented by nuclear spins, and adjacent spins can be distinguished. It is a further object of the present invention to cancel the spurious couplings within the crystal to reduce the error rate of the quantum computer.

The quantum computer of the present invention has the advantages that it has a relatively simple design, it can be cooled to low temperatures, and can be scaled up to include a large number of quantum bits.

Furthermore, the nuclear spins that store the quantum bits have a long relaxation time and are isolated from their surrounding environment. The nuclei are regularly spaced in the crystal lattice, allowing them to be precisely addressed. The nuclei have spin-1/2, and have no isotopes with different nuclear spins. Finally, the nuclei are surrounded by localized electrons, allowing a magnetic order of the electrons to be utilized to initialize the nuclear spins efficiently.

SUMMARY

A quantum computer comprises a crystal lattice having storage atoms. The storage atoms have nuclear storage spins. Quantum bits are stored as orientations of the storage spins. A magnetic field is applied to the crystal, the magnetic field having a gradient strong enough to cause energy levels of adjacent storage atoms along the direction of the gradient to be significantly different. The gradient is preferably created by a micromagnet in the vicinity of the crystal. The micromagnet preferably has a width on the order of 1 &mgr;m or less and generates a gradient on the order of 1 &mgr;m. The micromagnet preferably comprises dysprosium.

The electrons of the crystal acquire a regular order. This order is utilized to initialize the storage spins as follows: combined electron-nucleus transitions are induced in the crystal, resulting in the transfer of the electronic order to the polarizations of the storage spins.

The storage spins are decoupled from each other by applying a time-varying decoupling magnetic field. The decoupling field eliminates spurious couplings that otherwise plague the crystal. The decoupling field rotates the storage spins in such a way as to cancel their magnetic dipole-dipole interactions. The decoupling field preferably comprises series of &pgr;/2-pulses.

Quantum logic operations are performed on the storage spins. In a one-bit gate, a storage spin is rotated by an angle &thgr;. A two-bit exclusive-OR gate involves the interaction of two storage spins. To carry out the exclusive-OR operation, the two storage spins must be recoupled by either modifying or switching off the decoupling field. The exclusive-OR operation is then achieved by applying an oscillating magnetic field to the crystal, wherein the oscillating magnetic field has a frequency equal to a resonance frequency of the recoupled storage spins.

Measurement of final polarizations of the storage spins is achieved by applying an oscillating measurement magnetic field to the crystal and measuring an amount of absorption of the measurement field. In the preferred embodiment, the crystal comprises measurement atoms, and the measurement field has a frequency equal to a resonant frequency of a coupled system comprising at least one measurement atom and at least one storage atom.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1

shows a crystal used in a quantum computer.

FIG. 2

is an energy level diagram for electrons in the crystal.

FIG. 3A

illustrates the quantum computer according to a preferred embodiment of the invention.

FIG. 3B

shows a sample holder and a coil enclosing the crystal.

FIG. 3C

shows an assembly comprising the crystal.

FIG. 4

is an energy level diagram for a combined nucleus-electron system of the crystal.

FIGS. 5A and 5B

illustrate pulse sequences used in a decoupling magnetic field.

FIG. 5C

illustrates a pulse sequence used in a recoupling magnetic field.

FIG. 6

is an energy level diagram of a coupled system of two storage spins belonging to the crystal.

DETAILED DESCRIPTION

I. Crystal

FIG. 1

shows a crystal

10

used in a quantum computer of the present invention. Crystal

10

comprises spectator atoms

14

and active atoms having non-zero nuclear spins. The active atoms comprise storage atoms

12

. In the preferred embodiment, the active atoms also comprise measurement atoms

16

. Measurement atoms

16

are preferably identical to storage atoms

12

; the measurement atoms are distinguished from the storage atoms by their locations. This distinction is further discussed below.

For the purposes of illustration, only a few atoms appear in FIG.

1

. However, crystal

10

in general comprises any number of atoms, such as one hundred million or more.

The atoms of crystal

10

preferably form a regular lattice. The nuclei of storage atoms

12

have storage spins

24

, and the nuclei of measurement atoms

16

have measurement spins

26

. Quantum bits, or qubits, are stored by storage spins

24

. In the preferred embodiment, storage spins

24

are spin-1/2, and storage atoms

12

do not have isotopes with other nuclear spins. Only five elements,

19

F,

31

P,

89

Y,

103

Rh, and

169

Tm, satisfy these requirements. Therefore, in the preferred embodiment, storage atoms

12

and measurement atoms

16

are all either fluorine, phosphorus, yttrium, rhodium, or thulium.

In the preferred embodiment, storage atoms

12

and measurement atoms

16

form a salt structure with spectator atoms

14

. It is known that rare-earth elements form compounds known as “rare-earth salts” with group VA elements, including phosphorous. In the embodiments where the measurement atoms are phosphorous, it is preferred that the spectator atoms are atoms of a rare-earth element. Among the rare-earth elements, cerium is the simplest, since a cerium atom has zero nuclear spin (all other rare-earth elements have nonzero nuclear spins).

Therefore, in the preferred embodiment, storage atoms

12

and measurement atoms

16

are phosphorous, and spectator atoms

14

are cerium. In a cerium monophosphide crystal, each cerium atom has only one unpaired 4f electron. Crystals of cerium monophosphide, or CeP, are now obtainable using recently developed crystal growth techniques; see Y. S. Kwon et al., Physica B 171 (1991) p. 324 and Y. Haga et al., Physica B 206 & 207 (1995) p. 792.

Cartesian x, y, and z axes are oriented as shown in

FIG. 1

, preferably with each of the axes normal to a face of crystal

10

. The z-axis points in a longitudinal direction, and the x- and y-axes point in lateral directions perpendicular to the longitudinal direction.

A static magnetic field

20

is applied to crystal

10

. The static magnetic field has a component in the z-direction. The crystal is preferably kept at a temperature below the Nél temperature, approximately 8.5 K for CeP. In the preferred embodiment, when static magnetic field

20

is applied, a magnetic order is created among the electrons of spectator atoms

14

.

In the preferred embodiment, the spectator atoms are cerium atoms, and each cerium atom has a single unpaired 4f electron. The electron belongs to a

2

F multiplet, which is split by the spin-orbit interaction to produce a

2

F

7/2

and a

2

F

5/2

multiplet, see FIG.

2

. The

2

F

7/2

and

2

F

5/2

multiplets have an energy difference of approximately 280 meV, large enough that the higher energy

1

F

7/2

levels can be ignored.

A crystalline field present within crystal

10

splits the

2

F

5/2

manifold into a &Ggr;

8

quartet and a &Ggr;

7

doublet, as shown in FIG.

2

. The energy difference between the &Ggr;

8

and &Ggr;

7

multiplets is approximately 15 meV, an energy corresponding to a temperature of 170 K. Because crystal

10

is kept at a low temperature (preferably less than 8.5 K), the &Ggr;

8

quartet can be ignored.

Therefore each cerium atom of crystal

10

possesses an electron in the &Ggr;

7

doublet, the eigenstates of which are written |+> and |−>. The eigenstates |+> and |−> can be regarded as the eigenstates of an effective spin operator. Therefore each cerium atom has an effective spin

22

, as shown in FIG.

1

. Each of the effective spins is “up” or “down” (illustrated by a solid arrow pointing up or down in

FIG. 1

) depending on whether the corresponding electron is in the |+> state or the |−> state, respectively.

To reduce thermal excitations and thereby improve signal to noise in the quantum computer, crystal

10

is preferably kept at a temperature below 2 K. Typically, crystal

10

has a temperature of 1 K.

When static magnetic field

20

has a magnitude on the order of 1 T or less, effective spins

22

of the cerium atoms align in alternate &agr; and &bgr; layers, as shown in FIG.

1

. The layers are parallel to the xy-plane. In each a layer, the effective spins are up, and in each &bgr; layer, the effective spins are down.

When static magnetic field

20

is applied, the nuclear spins of the active atoms (that is, storage spins

24

and measurement spins

26

) couple to static magnetic field

20

via the Zeeman effect. The nuclear spins also couple to the effective spins of the neighboring electrons via the hyperfine interaction.

Static magnetic field

20

has a magnitude in the z-direction equal to B

z

(x,y). When B

z

(x,y)<1 T, the effective spins acquire the anti-ferromagnetic order illustrated in

FIG. 1

, and the nuclear spins of the active atoms are described by two different Hamiltonians, one for the a layers and one for the &bgr; layers.

In each &agr; layer, the storage spin

24

located at position (x,y) has a z-component I

&agr;

z

and an energy given by the following Hamiltonian:

H

&agr;

−(

g

I

&mgr;

N

B

z

(x,y)−A

hf

iso

−4

A

hf

)I

&agr;

z

  (1)

The first term in Eq. (1) gives the Zeeman splitting, the second gives the hyperfine splitting. The storage spin z-component I

&agr;

z

can take on values of +1/2 (up nuclear spin) or −1/2 (down nuclear spin). The constant g

i

is the g-factor for the active atoms; in the preferred embodiment the active atoms are phosphorous and g

I

=1.13. The quantity &mgr;

N

is the nuclear magneton. The quantity A

hf

iso

is the magnitude of an isotropic coupling, and has an experimentally determined value of approximately 4 MHz. The quantity A

hf

determines the strength of the hyperfine interaction, and is given by:

A

hf

=−(&mgr;

0

/4&pgr;)g

I

&mgr;

N

g&mgr;

B

/a

3

In this expression &mgr;

0

is the permeability of free space; g-factor for the effective spins and is equal to −10/7; &mgr;

B

is the Bohr magneton; and &agr; is a lattice spacing of crystal

10

, as shown in FIG.

1

.

In each &bgr; layer, the measurement spin

26

located at position (x,y) has a z-component I

&bgr;

z

and an energy given by:

H

&bgr;

=−(

g

I

&mgr;

N

B

z

(x,y)+

A

hf

iso

+4

A

hf

)I

&bgr;

z

  (2)

Eqs. (1) and (2) are similar, except for the signs in front of 4 A

hf

and A

hf

iso

, corresponding to opposite orientations of neighboring effective spins.

In the preferred embodiment, the active atoms in the a layers are storage atoms

12

, and the active atoms in the &agr; layers are measurement atoms

16

. More generally, it is preferred that the storage atoms are a collection of atoms that have identical hyperfine energies.

In a second embodiment, the active atoms in the a layers are the measurement atoms, and the active atoms in the &bgr; layers are the storage atoms. In a third embodiment, all of the active atoms are storage atoms, used for the storage and manipulation of quantum bits, regardless of their hyperfine energies. In this third embodiment, there are no measurement atoms.

When the strength of static magnetic field

20

is larger than 1 T, the effective spins

22

in crystal

10

align themselves in a manner more complicated than the alternating sequence of FIG.

1

. For example, the effective spins may form several spin-up layers in a row, followed by a number of alternating spin-up and spin-down layers. In the case of this and other complex magnetic order, it is still preferred that the storage atoms all have the same hyperfine energies.

Although it is presently preferred that crystal

10

comprises CeP, other compositions of crystal

10

are possible. In one embodiment, crystal

10

comprises CaF

2

. This choice is advantageous because most calcium atoms have no nuclear spin, and only a small percentage—0.13%—have spin 7/2. In another embodiment, crystal

10

comprises MnF

2

. The MnF

2

crystal establishes an anti-ferromagnetically ordered spin structure that has been well studied. In yet another embodiment, crystal

10

comprises CeF

3

. In these three embodiments, the storage atoms are fluorine atoms. For more information on materials to be used for crystal

10

, see Thaddeus Ladd et al., “Decoherence in Candidate Materials for Crystal Lattice Quantum Computation,” to be published in Applied Physics A. More information on CeP can be found in F. Yamaguchi et al., “Crystal Lattice Quantum Computer,” Applied Physics A 68 (1999) 1-8.

II. Static Magnetic Field

Static magnetic field

20

points primarily in the z-direction. Static magnetic field

20

furthermore has a gradient

21

in the x-direction, as illustrated in FIG.

1

. Gradient

21

gives storage atoms

12

different nuclear magnetic resonance frequencies, each resonance frequency depending upon the x-coordinate of the associated storage atom.

In some embodiments, gradient

21

has a component in the x-direction but not in the y-direction. In these embodiments, B

z

(x,y) is a function only of x, and is approximately uniform in planes parallel to the yz-plane within crystal

10

. In other embodiments, the gradient

21

has components both the x-direction and in the y-direction. In these embodiments, B

z

(x,y) is a function of both x and y.

In general, gradient

21

points in any direction, including the z-direction. In the presently preferred embodiments, gradient

21

lies in the xy-plane, and the present discussion assumes that gradient

21

is perpendicular to the z-axis. This assumption, however, should not be construed as a limitation of the invention.

FIG. 3A

illustrates a cutaway view of the quantum computer according to the preferred embodiment of the invention. A background magnetic field is generated by a magnet

34

. Magnet

34

is preferably a superconducting magnet. A first coil

60

resides within magnet

34

. First coil

60

is connected to a first power supply

33

.

Within first coil

60

is a second coil

62

which in turn encloses a sample holder

64

, as shown in FIG.

3

B. Second coil

62

is electrically connected to a second power supply

36

and to a signal processor

38

. The first and second coils provide time-varying magnetic fields to crystal

10

, as explained in detail below.

Inside sample holder

64

is an assembly

70

, as shown in FIG.

3

C. Assembly

70

comprises a micromagnet

30

attached to a substrate

32

. Micromagnet

30

has a coating

31

, to which is attached crystal

10

.

Micromagnet

30

is preferably ferromagnetic, and establishes a spatially varying magnetic field, thereby providing gradient

21

. Magnetic field

20

is the sum of the magnetic field due to micromagnet

30

and the background field created by magnet

34

.

Gradient

21

is preferably on the order of or greater than 1 T/&mgr;m. This magnitude gives a difference in B

z

(x,y) of roughly 10

−3

T between two adjacent storage atoms separated by 2a (approximately 12 angstroms). This difference corresponds to a Zeeman frequency difference of 8 kHz.

For gradient

21

to have a value of 1 T/&mgr;m or greater, micromagnet

30

must have a width w that is extremely small. In the preferred embodiment, width w is on the order of or less than 1 &mgr;m. In some embodiments, width w is even smaller, taking on values on the order of 0.1 &mgr;m or even 0.01 &mgr;m.

To fabricate micromagnet

30

with the desired width, micromachining and deposition techniques are used. In the preferred embodiment, micromagnet

30

comprises dysprosium. In another embodiment, micromagnet

30

comprises cobalt. Coating

31

optimizes gradient

21

within crystal

10

by giving a spacing between micromagnet

30

and crystal

10

. Coating

31

is preferably on the order of 1 &mgr;m thick. Coating

31

preferably comprises a polyamide.

Assembly

70

causes gradient

21

to point in the x-direction. In another embodiment, assembly

70

comprises two side-by-side micromagnets in a “split gate” configuration. This configuration causes magnetic field

20

to vary in both the x- and y-directions. In alternate embodiments, gradient

21

is established by one or more current-carrying electrodes. More information on micromagnet

30

can be found in J. R. Goldman et al., “Magnet Designs for a Crystal Lattice Quantum Computer,” to be published in Applied Physics A.

III. Initialization

The qubits of the quantum computer are initialized by giving storage spins

24

initial polarizations. The initialization takes advantage of the Pound-Overhauser double resonance on the anti-ferromagnetically ordered effective spins and the nuclear spins. Storage spins

24

and measurement spins

26

are initialized by inducing combined electron-nucleus transitions in crystal

10

.

A total magnetic field in the &agr; layers, B

&agr;

tot

(x,y) is the sum of static magnetic field

20

and an exchange magnetic field B

exch

generated by the neighboring effective spins:

B

&agr;

tot

(

x,y

)=B

z

(

x,y

)+

B

exch

.  (3)

Because of the anti-ferromagnetic structure of crystal

10

, the &agr; layers experience an exchange magnetic field in the opposite direction to that found in the &agr; layers. Therefore a total magnetic field in the &bgr; layers B

&bgr;

tot

(x,y) is:

B

&bgr;

tot

(

x,y

)=

B

z

(x,y)−

B

exch

.  (4)

The magnitude of B

exch

is on the order of 10T.

As shown in

FIG. 1

, a storage spin at position (x,y,z) is surrounded by six effective spins

22

, four of which are in the same xy-plane and two of which are located above and below the storage spin at locations (x,y,z±a). Transition energies for the effective spins in the &agr; layers is significantly different from transition energies for the effective spins in the &bgr; layers, primarily due to the different exchange magnetic fields, as given in Eqs. (3) and (4). Therefore, in considering the coupling of the storage spin at (x,y,z) to the effective spins in the P layers, the effective spins in the a layers can be ignored.

FIG. 4

shows an energy level diagram for the storage spin at position (x,y,z) and the effective spin immediately above the storage spin. The storage spin has z-component I

&agr;

z

, and the effective spin has z-component S

&bgr;

z

. In a state A, both I

&agr;

z

and S

&bgr;

z

equal+1/2; that is, both spins are up. In a state B, (I

&agr;

z

,S

&bgr;

z

)=(+1/2,−1/2). States C and D are characterized by (I

60

z

,S

&bgr;

z

)=(−1/2,+1/2) and (I

&agr;

z

,S

&bgr;

z

)=(−1/2,−1/2), respectively.

Transitions between states A and B, or between states C and D, are electron transitions. A combined electron-nucleus spin-flip transition

42

exists between states B and C. A transition between states A and D is not allowed.

At low temperatures, states B and D have roughly equal probabilities of being populated, and states A and C are unpopulated. To initialize the storage spin at (x,y,z), an oscillating magnetic field Bt having a frequency (i) is applied to crystal

10

, wherein frequency &ohgr;

1

has a value necessary to induce combined electron-nucleus transition

42

from state B to state C. When the transition is induced, state C decays quickly to state D, leaving I

&agr;

z

in the down state.

Frequency &ohgr;

1

is given by:

&ohgr;

1

=[|g|&mgr;

B

(

B

exch

−B

z

(

x,y

))+

g

I

&mgr;

N

B

z

(

x,y

)−(3/2)

A

hf

iso

−3

A

hf

]

.  (5)

In the above analysis, only the effective spin immediately above the storage spin at (x,y,z) was considered. An analysis of the coupling to the effective spin immediately below (x,y,z) shows that the oscillating magnetic field of frequency &ohgr;

1

can induce a combined electron-nucleus transition involving the effective spin below (x,y,z), again resulting in the storage spin being polarized in the down state.

Because frequency &ohgr;

1

depends upon B

z

(x,y), frequency &ohgr;

1

in general depends upon x and y. A plurality of oscillating magnetic fields having frequencies tuned according to the (x,y) locations of the storage atoms are applied. The oscillating magnetic fields cause the storage atoms and spectator atoms undergo combined electron-nucleus transitions that place substantially all of the storage spins into the down state.

Similarly, the measurement atoms at (x,y) are polarized by applying an oscillating magnetic field B

t2

having frequency &ohgr;

2

, wherein

&ohgr;

2

=[|g|&mgr;

B

(

B

exch

+B

z

(

x,y

))−

g

I

&mgr;

N

B

z

(

x,y

)−(3/2)

A

hf

iso

−3

A

hf

]

.  (6)

The result of applying B

t2

is to polarize the measurement atoms at (x,y) in the up state. Oscillating magnetic fields B

t

and B

t2

are generated by first coil

60

in FIG.

3

A.

In general, the storage spins are given initial directions, or polarizations, by inducing combined electron-nucleus transitions in crystal

10

. This is true even when the electronic structure is different from the anti-ferromagnetic structure of the effective spins illustrated in FIG.

1

. In some embodiments, the combined electron-nucleus transitions occur at optical frequencies.

IV. Decoupling

Storage spins

24

are grouped into collections, wherein all of the storage spins in a given computation collection experience the same magnetic field B

z

(x,y). That is, magnetic field

20

has the same strength at each storage spin

24

in a given collection.

For example, when gradient

21

lies in the x-direction, the storage atoms in each collection lie in a plane parallel to the yz-plane. When Bz(x,y) depends on both x and y, then each collection comprises storage atoms that lie in a line parallel to the z-axis.

A time-varying decoupling magnetic field is applied to crystal

10

to decouple the storage spins from each other. The purpose of the decoupling field is twofold. First, the decoupling field eliminates interactions between storage spins within the same collection. This allows the storage spins within each collection to act independently, and results in multiple parallel quantum calculations that do not interfere with each other.

The second purpose of the decoupling field is to decouple the different collections from each other. That is, the decoupling field also decouples storage spins having different locations along the direction of gradient

21

. This prevents the qubits represented by the storage spins in different collections from interfering with each other. When a quantum logic operation is to be performed on more than one collection, the relevant collections are recoupled during the operation. Quantum logic operations are further discussed below.

For the purposes of illustration, three storage spins are considered, and gradient

21

is parallel to the x-axis. A first storage spin I is the nuclear spin of the storage atom at coordinates (x

1

,y

1

,z

1

). A second storage spin I

2

is the nuclear spin of a storage atom at coordinates (x

1

+

2

a

,y

l

,z

l

). A third storage spin I

1

′ is the nuclear spin of a storage atom at coordinates (x

1

,y

1

,z

1

+

2

a

). In this illustration, spins I

1

and I

1

′ belong to the same collection; spins I

1

and I

2

belong to different collections.

In the preferred embodiment, the decoupling magnetic field comprises a first series of pulses

50

and a second series of pulses

52

, as shown in

FIGS. 5A and 5B

. P

x

and P

y

are &pgr;/2 pulses about the x- and y-axes, respectively. {right arrow over (P)}

x

and {right arrow over (P)}

y

are &pgr;/2 pulses about the negative x- and negative y-axes, respectively. A time interval T characterizes the timing of the pulses. The pulses separated by T or by 2T, as shown in

FIG. 5A. T

is typically on the order of 10 &mgr;sec.

First series

50

comprises &pgr;/2 pulses adjusted to operate on all storage spins having the same x-coordinate as I

1

. By applying the first series to crystal

10

, spins I

1

and

1

′ are decoupled. This decoupling can be demonstrated by noting that the interaction Hamiltonian between spins I

1

and I

1

′ is proportional to 3 I

1z

I

1

′

z

−I

1

·I

1

′, the average of which is zero under the influence of the first series.

Second series

52

is shown in FIG.

5

B. The second series is a series of &pgr;/2 pulses, the duration and frequency of which have been adjusted to operate on storage spins having x-coordinate equal to x

1

l +

2

a

. The interaction Hamiltonian between I

1

and I

2

is proportional to I

1z

I

2z

. Under the influence of series

50

and series

52

, the average interaction between I

1

and I

2

is zero. Therefore the decoupling field comprising the first and second series decouples the storage spins I

1

, I

1

′, and I

2

. In the preferred embodiment, crystal

10

comprises measurement atoms

16

, and the decoupling magnetic field also comprises components that decouple the measurement atoms from the storage atoms and from each other.

If it is desired to carry out a quantum logic operation on storage spins I

1

and I

2

, the spins are recoupled. This recoupling is accomplished by applying a recoupling magnetic field to crystal

10

. The recoupling field comprises first series

50

and a third series of pulses

56

, illustrated in FIG.

5

C. The recoupling field maintains the decoupling of spins I

1

and I

1

′ within the same collection, but couples the spin I

1

with the adjacent spin I

2

.

Other decoupling and recoupling schemes are used in other embodiments of the invention. In one alternate embodiment, the decoupling magnetic field comprises a sequence of &pgr;-pulses to spin I

1

, flipping the spin back and forth rapidly enough that it has no time-averaged effect on spin I

2

.

In another alternate embodiment, the decoupling magnetic field comprises a sequence of two alternating magnetic fields parallel to the x-axis, each applied for a time T

1

. The first alternating field rotates the storage spins located at (x,y) by an angle &thgr;

1

(x,y). The second alternating field differs from the first by a phase of &pgr;, and rotates the storage spins at (x,y) back to their original positions. Because static magnetic field

20

varies with x, angle &thgr;

1

(x,y) varies with x, and the average interaction between I

1

and I

2

is zero. In the alternate embodiments above, the spins I

1

and

2

are recoupled by temporarily switching off the decoupling magnetic field.

V. Quantum Logic Operations

All quantum logic operations on an arbitrarily large number of qubits can be built up from one-bit gates and two-bit exclusive-OR gates. These operations can be performed as follows.

(a) One-Bit Gate

An i

th

storage spin I

i

is rotated by an angle &thgr;. To accomplish this rotation, an alternating magnetic field B

q

is applied along the x-axis for a time duration &tgr; given by: (q

1

&mgr;

N

B

q

)&tgr;=&thgr;. The magnetic field B

q

has a frequency &ohgr;

q

chosen to satisfy the resonance condition determined by Eq. (1).

(b) Exclusive-OR Gate

A two-bit exclusive-OR operation is performed on i

th

storage spin I

i

and an adjacent storage spin I

i+1

, located a distance

2

a

along the x-axis from storage spin I

i

. The exclusive-OR operation flips spin I

1

only when I

i+1

has a predetermined value, or flips I

i+1

only when I

i

has a predetermined value.

The z-components of the storage spins I

i

and I

i+1

are m

i

and m

i+1

, respectively. Neglecting interactions with other storage spins, the energy of storage spin I

i

is: E

i

=

&ohgr;

i

m

i

, where &ohgr;

i

can be calculated from Eq. (1). Storage spins I

i

and I

i+1

are recoupled, either by using the recoupling field as outlined above, or by shutting off the components of the decoupling field that decouple spins I

i

and I

i+1

. The storage spins then form a coupled system with energy levels given by:

E=

&ohgr;

i

m

i

+

&ohgr;

i+1

m

i+1

+A

nd

1

m

i

m

i+1

  (7)

In Eq. (7), the term A

nd

1

gives a nuclear dipole interaction between storage spins I

i

and

i+1

. In the preferred embodiment, where crystal

10

comprises CeP, A

nd

1

can be calculated from A

nd

1

=(&mgr;

0

/4&pgr;) (g

I

&mgr;

N

)

2

/(2a)

3

, and has a value of 120 Hz.

The coupled system described by Eq. (7) is illustrated in FIG.

6

. States G, H, I, and J correspond to (m

i

, m

i+1

)=(−1/2,+1/2), (+1/2,+1/2), (−1/2,−1/2), and (+1/2,−1/2), respectively. The energies of the states can be computed from Eq. (7). The corresponding resonant frequencies for transitions of the coupled system are given in FIG.

6

.

To operate the exclusive-OR gate, an oscillating magnetic field B

o

is applied for a length of time suitable to provide a &pgr;-pulse. B

o

has a frequency equal to one of the resonant frequencies for a single spin-flip transition in the system of FIG.

6

.

For example, when a &pgr;-pulse at frequency &ohgr;

i

+A

nd

1

/2

is applied, spin I

i

flips only when m

i+1

=+1/2. A &pgr;-pulse at frequency &ohgr;

i

−A

nd

1

2

flips spin I

i

only when m

i+1

=−1/2. Similarly, &pgr;-pulses at frequencies &ohgr;

i+1

;±A

nd

1

2

conditionally flip I

i+1

.

(c) Other operations

Other quantum logic operations are possible. Two qubits can be swapped as follows. In the coupled system of I

i

and I

i+1

represented in

FIG. 6

, a &pgr;-pulse is applied at frequency &ohgr;

i

−&ohgr;

i+1

. After this pulse is applied, m

i

is equal to the value that m

i+1

had before the pulse; similarly m

i+1

is equal to the prior value of m

i

.

Another quantum logic operation is a three-bit exclusive-OR gate. Storage spins I

i−1

, I

i

, and I

i+1

are three consecutive storage spins that lie in a line parallel to the x-axis. The three storage spins U

i−1

, I

i

, and I

i+1

are coupled together to form a three-spin system. The three-spin system has energies:

E=

&ohgr;

i−1