DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0055] In the following, the basic principles of the invention are outlined in an introduction, followed by a discussion of the implementation of quantum logic networks on cluster states and a discussion of the construction of quantum logic networks in practice.

[0056] In this patent application a new scheme for quantum computing is proposed based on one-qubit measurements on a pre-entangled resource state, such as the so-called “cluster states” introduced by Briegel and Raussendorf (Phys. Rev. Lett. 86, 910, 2001; E-print quant-ph/0004051 at <xxx.lan1.gov>).

[0057] The scheme is different from quantum logic networks, but each quantum logic network can be implemented on a cluster state.

[0058] Referring now to the figures of the drawings in detail and first, particularly, to FIG. 1 thereof, there is shown an example for a cluster state. The qubits ⊕ are randomly spread over the lattice. Empty sites are marked by a dot. Qubits which can be connected via a series of next-neighboring qubits belong to the same cluster. For example, the qubit in the right upper corner does not belong to the cluster formed by the other shown qubits. As explained in more detail below, cluster state can be produced by initially preparing all qubits in the state |+> and acting upon each pair of neighboring qubits by a conditional phase gate.

[0059] The advantage of quantum computing with a resource state in comparison with a quantum logic network is that, during the course of the information processing, only simple one-qubit measurements are needed and no further entanglement needs to be created for performing quantum logic gates. All the entanglement that is needed in the course of a quantum computation is provided by the initial resource state. In quantum logic networks the situation is different; here, entanglement is created during the course of the computation by the use of quantum logic gates. But the gates will typically be much harder to implement than 1-qubit measurements.

[0060] The cluster states form, in a sense, the substrate for arbitrary quantum logic networks. The circuit representing the logic network is imprinted on the cluster state by performing one-qubit measurements on a subset of the entangled qubits in the appropriate bases.

[0061] The invention is particularly suited for implementation in optical lattices storing ultra-cold atoms (see Jaksch et al., Phys. Rev. Lett. 82, 1975, 1999) and similar systems. An essential advantage of the invention is given by the fact that the cluster states which form the substrate for the quantum computer can be made very large. Thus, in this proposal for quantum computing one is provided with a scalable system. A cluster state can serve for a computational task bounded in its complexity by the size of the cluster, but otherwise freely choosable. Furthermore, the present invention supports the view of regarding entanglement as a resource for quantum computing. This is because the quantum state is destroyed in the process of imprinting the quantum logic network on it and can therefore not be used a second time.

[0062] In the following the principle of realization of a quantum logic network on a cluster state is outlined. FIG. 2 shows the schematics of a quantum logic network. As will be outlined in more detail below, quantum logic networks usually involve the creation of the desired input state out of a fixed state by quantum logic network 1 and performing the computation with that input state by a quantum logic network 2.

[0063] Furthermore, the construction of quantum logic networks on cluster states is discussed in detail. It is displayed how the CNOT gate and arbitrary one-qubit rotations can be implemented on a cluster state. These gates form a universal set, i.e. every quantum logic gate can be constructed out of these elementary parts. That the elementary gates can in the proposed implementation on cluster states be combined to form a quantum logic network is also shown. Furthermore, it is discussed how to eliminate qubits that are in the cluster state but are not required for the quantum logic network. It is also shown that the gates are to an extent allowed to vary in shape. Thus, a hole in the cluster, i.e. an unoccupied qubit site, does not matter since its role can be taken over by a neighboring qubit.

[0064] At that point, the practically occurring case, a quantum logic network on a cluster state with irregular arrangement of qubits is reduced to the case where only the qubits necessary for the gate are present and arranged in some canonical order.

[0065] At that point, one is ready to prove the two essential points: the gates can be combined to form a quantum logic network and, the randomness of the results of measurements necessary to imprint the network on the cluster state can be accounted for.

[0066] Furthermore, some additional material of the proposed implementation is presented, for example, a “crossing of wires” in two dimensions. An implementation of a conditional phase gate with capability of freely adjusting the rotation angle is displayed. Further, an extension of the scheme for quantum computation on cluster states discussed so far is given. It picks up the question that is important for the practical implementation: “I can create big cluster states, but the computations I want to perform are also quite extensive. It turns out that my cluster states are just not big enough to serve the purpose. How can I get round this?” There is an answer to that question which will be given later on.

[0067] The remainder of the introduction is devoted to a review on cluster states and their entanglement properties and further to experimental techniques needed to create and manipulate cluster states. A practical implementation of the invention bases on experimental techniques which are known as such in quantum optics, in particular in the field of trapping and cooling of neutral atoms in magnetic traps and optical lattices. Details of these techniques are available in the literature.

[0068] Cluster states can be created by randomly filling a lattice of sites 1 $i=\sum _{l=1}{}^{d}n_li_l,$

[0069] with n _l denoting any natural number, d the dimension of the lattice and i _l the lattice vectors, with particles representing qubits. Each site carries either one or none qubit. All qubits are initially prepared in the state |+>=(|0>+|1>)/sqrt(2). Then between all pairs of next-neighboring qubits a conditional phase gate is performed. Let the left or the lower (or the fore in d=3) qubit be qubit c (central) and the other be qubit n (neighbor). Then the conditional phase gate performed with these two qubits is:

S _c→n (π)=|0> _c <0| _c {circle over (×)}σ _z ⁽ⁿ⁾ +|1> _c <1| _c {circle over (×)}1 ⁽ⁿ⁾ (1)

[0070] All the gates in each one of the directions are performed simultaneously such that there are only d entanglement operations no matter what the number of qubits is. The temporal order of the gates is of no concern for the produced state since the conditional phase gates all commute. Qubits that can be connected by a path of occupied next neighboring sites belong to the same cluster; in turn those qubits which cannot be connected that way belong to different clusters. The quantum state of the qubits on a lattice is a product state of cluster states. For precise definition of cluster states see (Briegel and Raussendorf, Phys. Rev. Lett. 86, 910, 2001; E-print quant-ph/0004051 at <xxx.lan1.gov>).

[0071] Cluster states |Φ> _c can be described by the following set of eigenvalue equations: 2 $\begin{array}{cc}{\sigma }_x^{\left(a\right)}\underset{i\in \mathrm{nbrs}\left(a\right)}{\Pi }{\sigma }_z^{\left(i\right)}{\Phi 〉}_c=\pm {\Phi 〉}_c& \left(2\right)\end{array}$

[0072] Where a is an arbitrary occupied lattice site and nbrs(a) is, for example, the set of all occupied next-neighboring sites of a. The eigenvalue tl is determined by which next-neighboring sites of a are occupied. The quantum correlations of the cluster state |Φ> _c that follow from (2) have been essential for proving that quantum gates can be realized in this proposal of a quantum computer.

[0073] For practical realization of the proposed invention, using e.g. atoms trapped in optical lattices, 4 major requirements must be fulfilled.

[0074] (i) The atoms trapped in the lattice must be cooled to their motional ground state. This has been achieved to a very good approximation (see Hamann et al., Phys. Rev. Lett. 80, 4149, 1998; Kerman et al., Phys. Rev. Lett. 81, 439, 2000).

[0075] (ii) The optical lattice must be filled with high filling factor (probability for a lattice site to be occupied by a single atom), but can be filled irregularly. This has been achieved (see DePue et al., Phys. Rev. Lett. 82, 2262, 1999).

[0076] (iii) A controlled simultaneous interaction between the atoms, which results in a conditional phase gate between all pairs of neighboring qubits, must be implemented. With the experience of state of the art atom interferometry this is feasible (see Jaksch et al., Phys. Rev. Lett. 82, 1975, 1999; Briegel et al., J. Mod. Opt. 47, 415, 2000).

[0077] (iv) The atoms have to be addressed individually to permit selective one-qubit measurements. To accomplish this, the scheme of cooling the atoms and then performing a controlled interaction between them must be extended (see Scheunemann et al., Phys. Rev. A 62, 051801(R), 2000; Weitz, IEEE J. Quant. Electronics 36, 1346, 2000).

[0078] With regard to item (iii) it is noted that the atoms are cooled and trapped by an electromagnetic field, supplied by two counterpropagating linearly polarized laser beams for each spatial direction. (The following is described in detail in Jaksch et al., Phys. Rev. Lett. 82, 1975, 1999; Briegel et al., J. Mod. Opt. 47, 415, 2000). For each of the spatial directions, the polarization directions of the two respective laser beams differ by an angle which can be controlled (lin-angle-lin configuration). If now the laser frequency is adjusted to a particular value the individual atoms are subject to a state dependent potential.

[0079] For both internal states of the atoms, |0> and |1>, the electric potential remains sinusoidal, but with the minima translated. The relative translation distance is controlled by the angle between the polarization directions of the laser beams. If this angle is varied adiabatically such that the atoms can follow the moving potential without being excited, the atoms on the lattice can be translated depending upon the state which they are in. If the relative translation between the two potentials seen by the atoms is equal to the distance between neighboring atoms, these neighboring atoms can be made to interact via s-wave scattering.

[0080] By this interaction a quantum gate between neighboring atoms is realized. After the interaction the atoms are brought back to their initial positions by turning the angle between the two polarization directions back to zero. It is important to notice that the interaction in this scheme is performed simultaneously between all pairs of neighboring atoms. If the atoms in the lattice are prepared in the quantum mechanical superposition |+>=(|0>+|1>)/sqrt(2) and the interaction time is adjusted properly then the emerging quantum state is the required cluster state. The initial superposition |+> for the internal state of each atom is achieved by applying a laser pulse of appropriate intensity and duration (π/2-pulse) on the whole sample of atoms.

[0081] The frequency of the counterpropagating lasers which constitute the trapping potential determines the distance between neighboring atoms. The drawback of the scheme is that to make the potential seen by the atoms state dependent, the frequency of the lasers must be high (e.g. at an optical frequency which is between the S1/2-P1/2 and S1/2-P3/2 transition line for Rb atoms) and, correspondingly, the distance between neighboring atoms on the lattice is very small. Therefore, individual atoms cannot be resolved by standard spectroscopic methods. In this scheme, measurements on individual qubits are not straightforwardly possible.

[0082] Accordingly, it is preferred that the lattices for cooling the atoms and for performing the interaction between neighboring atoms are different. As described in detail in Scheunemann et al., Phys. Rev. A 62, 051801(R), 2000; Weitz, IEEE J. Quant. Electronics 36, 1346, 2000, the wavelength of the infrared laser used for cooling and-initial trapping is a multiple of the wavelength of the laser used to accomplish the interaction. In the first step, the atoms are cooled and trapped by in an optical lattice created by the infrared lasers (Friebel et al., Phys. Rev. A 57, R20, 1998). In a second step, after cooling, the infrared lattice is switched off and the short wavelength lattice is switched on. Since the lattice constant of the former lattice is a multiple of the lattice constant of the latter, potential minima can be arranged to coincide. Thus, the atoms are adiabatically transferred without being excited. Now, the interaction is performed as before. The only difference is that the atoms have to be carried over a larger distance since the short wavelength lattice is now populated only on a sublattice. Now, the atoms are separated far enough from each other to permit individual addressing. One-qubit measurents are for example performed by means of conditional resonance fluorescence techniques, also referred to as quantum jump method (Cirac and Zoller, Phys. Rev. Lett. 74, 4091, 1995), which can be performed with near 100% detection efficiency (Nagourney et, al., Phys. Rev. Lett. 56, 2787, 1986; Sauter et al., Phys. Rev. Lett. 57, 1696, 1986; Bergquist et al., Phys. Rev. Lett. 57, 1699, 1986). This extended scheme is a subject of current experiments (see Weitz, “Towards controlling larger quantum systems: From laser cooling to quantum computing”, IEEE J. Quant. Electronics 36, 1346, 2000).

[0083] While we have described a particular implementation of the proposed method of quantum computation via one-qubit measurements, namely in optical lattices, there are other systems in which the method could be implemented. This includes arrays of magnetic microtraps on nanofabricated surfaces (Calarco et al., Phys. Rev. A 61 022304, 2000) and arrays of capacitively coupled quantum dots in semiconductor materials (Tanamoto, E-print quant-ph/0009030 at <xxx.lanl.gov>), for example. Similar ideas can be applied to system of Josephson junctions (J. E. Mooij et al., Science, 285, p. 1036 (1999)).

[0084] In the following the principles of quantum computing with entangled resource states are explained. FIG. 3 a shows an example for a realization of an elementary quantum computer on a cluster state. As will be explained in more detail below, an embodiment of the present invention involves the creation of the desired input state from a fixed state. Thereafter the computation evolves by imprinting the quantum logic network and reading out the results. Those steps are all performed only by one-qubit measurements. FIG. 3 b illustrates quantum computing by measuring two-state particles on a lattice. Before the measurements the qubits are in the cluster state as given by equation (2). Circles ⊙ symbolize measurements of σ _z , vertical arrows are measurements of σ _x while tilted arrows refer to measurements in the x-y plane.

[0085] Quantum computing with cluster states is a quantum computing scheme conceptually different from quantum logic networks. However, it will be shown that each quantum logic network can be implemented on a cluster state. Every computation that can be done with a quantum logic network can also be performed by the use of a cluster state.

[0086] As can be seen from FIG. 2 quantum computing based on a quantum-logic network involves the creation of the input state out of a fixed state, e.g. |+> ^N by a quantum logic network 1 and computation with that input state performed by quantum logic network 2. To accomplish this with the help of a cluster state, the following steps have to be performed:

[0087] reaction of the cluster state |Φ> _c

[0088] measuring all but the output qubits in the appropriate basis (programming the cluster state); and

[0089] measuring the output qubits (readout).

[0090] An example of imprinting a quantum logic gate on a cluster states is given in FIGS. 4 a and 4 b . In FIG. 4 a , a cluster of qubits is shown. Thereby, of those qubits characterized by a , σ _z is measured, of those qubits characterized by a {circle over (×)}, σ _x is measured, and those qubits characterized by a ◯ are the output qubits. Measurement outcomes (arbitrary choice shown) are designated by {0,1} for σ _z -measurements and {+,−} for σ _x -measurements, respectively. In FIG. 4 b , the corresponding quantum gate that is thereby realized (CNOT) and the corresponding output state are shown.

[0091] There is also a simplified scheme containing only one step. Normally, the input state of the qubits is a product state, i.e. a disentangled pure state. Then, step 1 in the above task-list can be omitted, if instead writing the input is included in the preparation of the initial state. Then, the input qubits are set to their required quantum state and all other qubits on the cluster are set to the individual state |1> as usual. Further, the usual entanglement step, the conditional phase gate between neighboring qubits, is performed. Now, only the quantum logic network to perform the calculation itself must be implemented by one-qubit measurements on the cluster, while the step imprinting the quantum logic network to set the input state drops out. This makes the implementation of the quantum logic network more compact, but on the other hand the simplified scheme is conceptually less powerful. Now one has to tailor a specific initial state to imprint the quantum logic network on the specific initially prepared state, while in the full scheme one uses a universally applicable resource, the cluster state. On the, other hand, in order to use the described quantum computer for quantum cryptography, unknown quantum states must be processed. This can be done with the described procedure where the group of particles representing the output register is not measured. In this way, an unknown quantum state is read in and processed. The output in this mode of running the quantum computer is a quantum state, carried by the set of quantum systems which form the output register. This output state can then e.g. be stored for later use, or teleported, or otherwise transmitted to any other participant in the cryptographic network. Such a processing is an important ingredient in quantum repeaters needed for long-distance quantum communication (Briegel et al., Phys. Rev. Lett. 81, 5932 (1998)).

[0092] In the following, it will be shown how to implement a universal set of quantum gates, the CNOT and arbitrary one-qubit rotations, and how to combine these building blocks to quantum logic networks. Especially, the implementation of the elementary piece of wire, arbitrary one-qubit rotations and the CNOT gate are addressed for the case that no other than the required qubits are present in the quantum state.

[0093] FIG. 5 shows an example for an implementation of a CNOT gate on a cluster state of 4 qubits. In the following, it is shown how a CNOT gate between two qubits, target qubit 1 and control qubit 4, is implemented on a cluster-like state of 4 qubits. The input state consists of qubit 1 (target) and 4 (control). Qubits 1 and 2 are measured in the x-basis to perform the gate. The output state is encoded in qubits 3 (target) and 4 (control). By the action of the gate the result of the computation is transferred to a qubit 3, while the control qubit remains at its site. Specifically, we want to implement the following gate:

CNOT (1→3,4)=|0><0| ₄ {circle over (×)}1 ^(1→3) +|1><1| ₄ {circle over (×)}σ _x ^1→3 (3)

where

σ _x ^(1→3) =|0> ₃ <1| ₁ +|1> ₃ <0| ₁

[0094] To implement the CNOT gate one proceeds as follows. Suppose it has been prepared the 4 qubit state:

|Ψ> _1-4 =|ψ ₁₄ > _1,4 {circle over (×)}|+> ₂ {circle over (×)}|+> ₃ (4)

[0095] where the input state has been encoded by |Ψ ₁₄ > _1-4 in the qubits 1 and 4. On the state |Ψ> _1-4 the entanglement operation S _1→2 {circle over (×)}S _2→3 {circle over (×)}S _4→2 is performed, which concludes the state preparation. If the qubits are arranged in the pattern displayed in FIG. 5 then this is the entanglement operation which is automatically carried out by the simultaneous conditional phase gate. To implement the gate, now measure the qubits 1 and 2 in the x-basis, i.e. 3 $\begin{array}{cc}\mathrm{qubit}1\mathrm{in}B_1=\left\{\frac{{{0〉}_1+1〉}_1}{\sqrt{2}},\frac{{{0〉}_1-1〉}_1}{\sqrt{2}}\right\}\mathrm{qubit}2\mathrm{in}B_2=\left\{\frac{{{0〉}_2+1〉}_2}{\sqrt{2}},\frac{{{0〉}_2-1〉}_2}{\sqrt{2}}\right\}& \left(5\right)\end{array}$

[0096] The output state then is

|μ> _1-4 =|s ₁ > ₁ {circle over (×)}|s ₂ > ₂ {circle over (×)}|ψ ₃₄ > _3,4′ with |ψ ₃₄ >=CNOT′|ψ ₁₄ > (6)

where

CNOT′=σ _z ^{(3) s ₁ +1} σ _x ^{(3) s2} σ _z ^{(4) s1} CNOT (1→3,4) (7)

[0097] The quantum computation is completed by the readout, i.e. one-qubit measurements on the output qubits. For an isolated gate, whose cluster state implementation to explain was the aim of this subsection, the extra rotations σ _x and 94 _z in equation (7) that are carried out in addition to the desired CNOT-gate can be compensated. Which extra rotations occurred is known from the measurement results s ₁ and s ₂ . To compensate for them, one applies the inverse rotations on the output before the measurements are performed. One can equivalently omit the counter-rotations, but then has to change readout measurement directions or interpretation of readout measurement outcomes, respectively. How the unwanted extra rotations can be accounted for if the gate is part of a quantum logic network will be explained later on.

[0098] An arbitrary one-qubit rotation U in SU(2) can be performed on a chain of five qubits, e.g. between a qubit 1 at site i and a qubit 5 at site i+4i ₁ . Initially, a quantum state

|Ψ> _1-5 =|ψ ₁ > ₁ {circle over (×)}|+> ₂ {circle over (×)}|+> ₃ {circle over (×)}|+> ₄ {circle over (×)}|+> ₅

[0099] is prepared and acted upon by the simultaneous entanglement operation S _1→2 {circle over (×)}S _2→3 {circle over (×)}S _3→4 {circle over (×)}S _4→5 . Let us assume for a moment that then qubits 1 to 4 are measured in the following bases 4 $\begin{array}{cc}\mathrm{qubit}1\mathrm{in}B_1=\left\{\frac{{{0〉}_1+1〉}_1}{\sqrt{2}},\frac{{{0〉}_1-1〉}_1}{\sqrt{2}}\right\}\mathrm{qubit}2\mathrm{in}B_2\left(\alpha \right)=\left\{\frac{{{0〉}_2+{}^{i\alpha }1〉}_2}{\sqrt{2}},\frac{{{0〉}_2-{}^{i\alpha }1〉}_2}{\sqrt{2}}\right\}\mathrm{qubit}3\mathrm{in}B_3\left(\beta \right)=\left\{\frac{{{0〉}_3+{}^{i\beta }1〉}_3}{\sqrt{2}},\frac{{{0〉}_3-{}^{i\beta }1〉}_3}{\sqrt{2}}\right\}\mathrm{qubit}4\mathrm{in}B_4\left(\gamma \right)=\left\{\frac{{{0〉}_4+{}^{i\gamma }1〉}_4}{\sqrt{2}},\frac{{{0〉}_4-{}^{i\gamma }1〉}_4}{\sqrt{2}}\right\}& \left(9\right)\end{array}$

[0100] where the respective qubits j, j=1 . . . 4, are projected into the former (latter) eigenstate of the measurement basis Bj if the measurement result s _j is 0 (1). The state |μ> _1-5 which by that emerges is

|μ> _1-5 =|s ₁ > ₁ {circle over (×)}|s ₂ > _2,α {circle over (×)}|s ₃ > _3,β {circle over (×)}|s ₄ > _4,γ {circle over (×)}|ψ ₅ > ₅ with |ψ ₅ >=U|ψ ₁ < (10)

[0101] Therein U is given by

U (α,β,γ)=σ _x ^{s ₂ +s ₄} σ _z ^{s ₁ +s ₃} U _x ((−1) ^{s ₁ +s ₃} γ) U _z ((−1) ^{s ₂} β) U _x ((−1) ^{1+s ₁} α) (11)

[0102] where U _x (α) is a rotation about the x axis about the angle α etc. In the standard basis {|0>, |1>} the rotations U _x and U _z read in matrix form 5 $\begin{array}{cc}\left(U_x\left(\alpha \right)\right)=\left(\begin{array}{cc}\cos \frac{\alpha }{2}& -i\sin \frac{\alpha }{2}\\ -i\sin \frac{\alpha }{2}& \cos \frac{\alpha }{2}\end{array}\right),\left(U_z\left(\beta \right)\right)=\left(\begin{array}{cc}{}^{-i\frac{\beta }{2}}& \\ & {}^{i\frac{\beta }{2}}\end{array}\right)& \left(12\right)\end{array}$

[0103] From equation (11) we see that procedure (9) yields a unitary operation composed of a sequence of rotations. The rotation angles of each of the first three rotations is determined by the measurement direction of one of the qubits and the outcomes of the measurements of some other qubits (sign of the rotation angle). In particular we note that whether the first rotation (U _x ) is about the angle α or −α is determined by the result s ₁ of the measurement of qubit 1. The angle α itself is adjusted by choosing the basis B ₂ (α) for measurement of qubit 2. In the same manner, whether the second rotation (U _z ) is about β or −β is determined by the measurement result at qubit 2, while the angle β itself is adjusted by choosing the measurement basis B ₃ (β) of qubit 3. Whether the third rotation (U _x ) is about the angle γ or −γ is determined by the measurement results at qubits 1 and 3 while the angle γ itself is adjusted by choosing the measurement basis B ₄ (γ) at qubit 4. From this observation it follows that in order to perform a rotation one has to perform the measurements at the qubits 1-4 in the order 1, 2, 3, 4.

[0104] Measurement bases have to be chosen according to previously obtained measurement results. The sequence of three rotations about angles ±α, ±β and ±γ is followed by possible special rotations σ _x and σ _z depending on the outcomes of the measurements at qubits 1 to 4, s ₁ , . . . , s ₄ . As will be shown, these additional rotations can be accounted for at the end of the computation. Note that the sequence of the first 3 rotations very much resembles the representation of an arbitrary rotation in terms of Euler angles.

[0105] We want to perform the rotation

U _R (ξ,η,ζ)= U _x (ζ) U _z (η) U _x (ξ) (13)

[0106] on a qubit where ξ, η and ζ are the Euler angles. To do that we perform the following steps (procedure 14):

[0107] (i) Measure qubit 1 in the basis B ₁ .

[0108] (ii) Depending on the measurement result at qubit 1, s ₁ =0 or s ₁ =1, measure qubit 2 in the basis B ₂ (−ξ) or B ₂ (ξ).

[0109] (iii) Depending on the measurement result at qubit 2, s ₂ =0 or s ₂ =1, measure qubit 3 in the basis B ₃ (η) or B ₃ (−η).

[0110] (iv) Depending on the measurement result at qubit 1 and 3, s ₁ +s ₃ mod2=0 or s ₁ +s ₃ mod2=1, measure qubit 4 in the basis B ₄ (ξ) or B4(−ξ)

[0111] By this procedure (14), the rotation U′ _R is achieved

U′ _R =σ _x ^{s ₂ +s ₄} σ _z ^{s ₁ +s ₃} U _R (15)

[0112] As already stated, in the following it will be explained -how the possible σ _x - and σ _z -rotations can be corrected at the end of the computation.

[0113] At this point we emphasize, that this procedure as it stands is only valid for an isolated one-qubit rotation. If the rotation becomes part of a quantum logic network the procedure will slightly change. This issue is discussed later on.

[0114] The elementary piece of wire takes the state |ψ ₁ > of a qubit two qubits further. It is important to note here that the standard piece of wire is not between neighboring qubits. Consider a chain of 3 qubits, where qubit 1 had been prepared in |ψ ₁ >, qubits 2,3 in |+> and where qubits 1 and 2 have been measured in ν _x , after the entanglement operation. With similar definitions as before:

|ψ ₃ >=U _wire |ψ ₁ > (16)

[0115] where we want

U _wire =1 (17)

[0116] By measuring qubits 1 and 2 in σ _x we obtain the following unitary operation U _wire , instead of U _wire :

U′ _wire =ν _x ^{1+s ₂} σ _z ^{1+s ₁} 1 (18)

[0117] and again, the possibly occurring rotations σ _x and σ _z can be corrected at the end of the computation.

[0118] It will occur very often that there are qubits in the cluster state which are not needed for the quantum logic network one wants to implement. Those qubits are measured in the σ _z -basis. After that, the superfluous qubits have only minor influence on the quantum logic network and this influence can be corrected; what needs to be done is to change the interpretation of measurement results on the essential qubits.

[0119] To see this we define the following sets of qubit sites

set of sites of qubits essential for the quantum logic network.

₀ set of sites of superfluous qubits which are projected into |0> by the σ _z -measurement.

(19)

₁ set of sites of superfluous qubits which are projected into |1> by the σ _z -measurement.

⊂ Set of sites of essential qubits which are affected by the superfluous qubits at sites in ₀ ∪ ₁

[0120] Further, we introduce the two functions ubrs(a) (upper neighbors) and lbrs(a) (lower neighbors).

ubrs(a) Set of all those next neighboring sites 6 $\stackrel{->}{j}=\sum _{l=1}^dj_l{\stackrel{->}{e}}_l\mathrm{of}\mathrm{site}\stackrel{->}{a},\mathrm{for}\mathrm{which}\sum _{l=1}^dj_l=\left(\sum _{l=1}^da_l\right)+1$ lbrs(a) Set of all those next neighboring sites ( 20 ) 7 $\begin{array}{cc}\stackrel{->}{j}=\sum _{l=1}^dj_l{\stackrel{->}{e}}_l\mathrm{of}\mathrm{site}\stackrel{->}{a},\mathrm{for}\mathrm{which}\sum _{l=1}^dj_l=\left(\sum _{l=1}^da_l\right)-1& \left(20\right)\end{array}$

[0121] With the definitions (20) one has ubrs(a)∪lbrs(a)=nbrs(a); for all a. Now be |Φ _E a cluster state of only essential qubits and |Ψ _E a state obtained from a cluster state |Φ> _C , C=E∪S ₀ ∪S ₁ where the superfluous qubits have been measured in σ _z . Then, the two states |Φ> _E and |Ψ> _E can be transformed into another by a local unitary transformation, 8 $\begin{array}{cc}{\Psi 〉}_{\varepsilon }=\left(\underset{i\in S_0}{\Pi }\underset{j\in \mathrm{ubrs}\left(i\right)}{\Pi }{\sigma }_z^{\left(j\right)}\right)\left(\underset{i^{\prime }\in S_1}{\Pi }\underset{j^{\prime }\in \mathrm{lbrs}\left(i^{\prime }\right)}{\Pi }{\sigma }_z^{\left(j^{\prime }\right)}\right){\Phi 〉}_{\varepsilon }& \left(21\right)\end{array}$

[0122] The affected essential qubits have their site in the set R, i.e. R consists of all those qubit sites on whose qubits in equation (21) an odd number (σ _z ² =1) of rotations σ _z is applied. 9 $\begin{array}{cc}{\Psi 〉}_{\varepsilon }=\underset{i\in R}{\Pi }{\sigma }_z^{\left(i\right)}{\Psi 〉}_{\varepsilon }& \left(22\right)\end{array}$

[0123] In all the discussed cases measurements of essential qubits have their measurement axis in the x/y-plane, i.e. σ _x cos φ+σ _y sin φ is measured. To see the effect on the interpretation of measurement outcomes we compare the states |Φ> _E and |Ψ> _E but with a further qubit at site k, k in R, measured. Be 10 $\begin{array}{cc}{P}_{\pm }^{\left(k\right)}=\frac{1\pm \left(\cos \varphi {\sigma }_x+\sin {\mathrm{\varphi \sigma }}_y\right)}{2}& \left(23\right)\end{array}$

[0124] the projector that describes the action of the measurement at -qubit site k on the states |Φ> _E and |Ψ> _E . Then, by equations 22 and 23, one finds 11 $\begin{array}{cc}{\left(P_{\pm }^{\left(k\right)}\Psi 〉\right)}_{\varepsilon }=\underset{i\in R}{\Pi }{\sigma }_z^{\left(i\right)}P_{\mp }^{\left(k\right)}{\Phi 〉}_{\varepsilon }& \left(24\right)\end{array}$

[0125] Thus, the meanings of results of measurements on a state |Ψ> _E at qubits with their sites in R are interchanged in comparison with the meanings of these measurement results for |Φ> _E .

[0126] To eliminate the effect of qubits which are present in the cluster but not necessary to implement the quantum logic network, it is preferred to

[0127] (i) measure the superfluous qubits in σ _z . Determine the sets s ₀ , s ₁ and from them the set R.

[0128] (ii) measure the essential qubits in the usual manner. For qubits at sites k in R keep the inverse s _k of the measurement result s _k , s _k =1−s _k . For qubits at sites k′ in E\R keep the measurement result s _k′ .

[0129] (iii) on the basis of the (modified) measurement results s _k′ ( s _k ) compute the counter-transformation to act upon the output state along the same lines as if there were no superfluous qubits present and no measurement results inverted. Perform readout measurements in the usual manner.

[0130] FIG. 6 shows an example for an implementation of a CNOT gate on a cluster state illustrating this procedure. To implement a CNOT between qubits 1 and 4, first the superfluous qubits 5, 6, 7 are eliminated by measuring them in σ _z . In this example E={1, 2, 3, 4}, s ₀ ={5}, s ₁ ={6, 7}, R={3, 4}. The state of qubits that emerges after the σ _z -measurements is local unitary equivalent to the state obtained after initial state preparation if the qubits 5, 6, 7 were not present. The local unitary transformation by which the two states can be transformed into each other is σ _z ⁽³⁾ σ _z ⁽⁴⁾ . Therefore, if qubits 3 and 4 are measured in σ _x , σ _y measurement results s _3/4 =0(1) are equivalent to s _3/4 =1(0) in the standard case (qubits 5, 6, 7 not present).

[0131] The shape of the gates and of wire is variable. Gates and wire can be rotated and bent, but there is one rule to respect: qubits which were not next neighbors in the standard implementation of the quantum logic network cannot become next neighbors in the implementation with bent gates and wire.

[0132] The variability of the shape of gates and wire is important since one has in general no control of the filling pattern of the qubit lattice. Qubit clusters that one is thus confined to work with may have “holes” at lattice sites where one wants to imprint a wire, and one therefore must be allowed to lay a bypass. The system is flexible enough to permit this compensation for the irregularity of the lattice filling, pattern.

[0133] The reason for this robustness is the following equation obeyed for the conditional phase gate defined in equation (1) which is simultaneously carried out between all pairs of neighboring qubits to produce the entangled initial state.

S _c→n (π)=σ _z ^(c) σ _z ⁽ⁿ⁾ S _n→c (π) (25)

[0134] If the shape of a gate is varied all that occurs are interchanges of the role of control and target qubits for some of the simultaneously performed conditional phase gates in the course of the resource creation. But this produces, in comparison to the output state of standard implementation, in accordance with equation (25) only extra rotations σ _z , on some qubits. Thus, the situation encountered here is the same as before where the influence of superfluous qubits was discussed. Let |Φ> be the cluster state only of essential qubits and them being in some standard arrangement; and |Ψ> the cluster state of the same number of qubits, but in different arrangement due to variation of the shape of some gates. Then the following equation holds 12 $\begin{array}{cc}\Psi 〉=\underset{k\in R_{\mathrm{vs}}}{\Pi }{\sigma }_z^{\left(k\right)}\Phi 〉& \left(26\right)\end{array}$

[0135] where the set R _vs is determined by use of equation (25). Therefore, before interpretation measurement results on qubits at sites k in R _vs must be inverted. To reduce a quantum computation on a cluster state with qubits in different arrangement as compared to some standard implementation proceed as follows:

[0136] (i) Measure the qubits implementing the quantum logic network in the appropriate bases and in the usual order.

[0137] (ii) From the shape of the cluster and by use of equation (25), determine the set R _vs of qubits sites upon whose qubits acted, beyond the simultaneous conditional phase gate, an extra rotation σ _z in comparison to the standard qubit arrangement.

[0138] (iii) Invert all measurement outcomes s _k for qubit sites k in R _vs . Then proceed as usual.

[0139] As an example, the reinterpretation of measurement results on a 5-qubit piece of wire in snake-like shape as compared to the same piece of wire in some standard (angle) arrangement is displayed in FIG. 7 .

[0140] As illustrated in FIG. 7 , the 5 qubits of this piece of wire are in the same order in both cases, but the shape of the right wire is varied as compared to the left one (in standard arrangement). Accordingly, for the initially applied conditional phase gates between qubits 2 and 3, and 4 and 3 respectively, control and target qubit are interchanged. After state preparation, the transformation that takes the 5-qubit state displayed at the right to the state displayed at the left is σ _z ^(2)σ _z ⁽⁴⁾ . As a consequence, when measuring σ _x of qubits 1 to 4 to implement a piece of wire, measurement results at sites 2 and 4 have to be inverted before being interpreted with rules for standard arrangement of qubits.

[0141] In the following the so-called “connectivity property” is discussed. What we termed “connectivity property” is the fact that the action of a quantum logic network can be understood from its elementary components. I.e. a quantum gate with “quantum” wires (by “quantum” wire we mean in this context a wire for qubit quantum states, not a carbon nano-tube or similar device for which the term is usually reserved) attached to its legs should have no other effect than the gate itself, and two pieces of “quantum” wires put together should act like one “quantum” wire. In general, we want to combine the elementary gates to quantum logic networks in the same manner as we combine classical gates and wire to classical logic networks.

[0142] To keep the notation transparent, for the moment we will confine ourselves to the proof in a very special case, the combination of two elementary pieces of wire to a wire that takes a qubit four lattice sites further. But this very special case already covers the essence of the argument.

[0143] The standard procedure to implement wire between qubits 1 and 5 is

1) prepare the state |ψ ₁ > ₁ {circle over (×)}|+> ₂ {circle over (×)}|+> ₃ {circle over (×)}|+> ₄ {circle over (×)}|+> ₅

2) apply standard entanglement procedure, the simultaneous conditional phase gates: S _1→2 (π){circle over (×)}S _2→3 (π){circle over (×)}S _3→4 (π){circle over (×)}S _4→5 (π). This concludes state preparation.

(27)

3) measure qubits 1 to 4 in σ _x i.e. apply the projector 13 $\underset{i=1}{\overset{4}{\Pi }}P_{x,s_i}^{\left(i\right)}\mathrm{with}P_{x,s_i}^{\left(i\right)}=\frac{1+{\left(-1\right)}^{s_i}{\sigma }_x}{2}$ to the previously prepared state.

[0144] By this procedure the state of a qubit initially stored at qubit site 1 is transferred to the qubit at site 5.

[0145] Now note that the measurements on qubits 1 and 2 commute with two of the entanglement operations S(π): 14 $\begin{array}{cc}\lfloor P_{x,s_1}^{\left(1\right)},S_{3->4}\left(\pi \right)\rfloor =0,\left[P_{x,\mathrm{s2}}^{\left(2\right)},S_{3->4}\left(\pi \right)\right]=0\left[P_{x,s_1}^{\left(1\right)},S_{4->5}\left(\pi \right)\right]=0,\left[P_{x,\mathrm{s2}}^{\left(2\right)},S_{4->5}\left(\pi \right)\right]=0& \left(28\right)\end{array}$

[0146] Thus, the procedure (27) to implement wire has the same effect as the following procedure:

[0147] 1) prepare the state |ψ ₁ > ₁ {circle over (×)}|+> ₂ {circle over (×)}|+> ₃ {circle over (×)}|+> ₄ {circle over (×)}|+> ₅

[0148] 2) apply a part of the standard entanglement procedure: S _1→2 (π){circle over (×)}S _2→3 (π)

[0149] 3) measure qubits 1 and 2 in σ _x i.e. apply the projector 15 $\begin{array}{cc}\underset{i=1}{\overset{2}{\Pi }}P_{x,s_i}^{\left(i\right)}& \left(29\right)\end{array}$

[0150] 4) apply the remaining part of the standard entanglement procedure: S _3→4 (π){circle over (×)}S _4→5 (π)

[0151] 5) measure qubits 3 and 4 in σ _x i.e. apply the projector 16 $\underset{i=3}{\overset{4}{\Pi }}P_{x,s_i}^{\left(i\right)}$

[0152] In procedure (29) after initial preparation of a product state steps 2 and 3 realize a wire between qubits 1 and 3. The state that emerges is |s ₁ > ₁ {circle over (×)}|s ₂ > ₂ {circle over (×)}|Ψ ₃ > ₃ {circle over (×)}|+> ₄ {circle over (×)}|+> ₅ with |Ψ ₁ > and |Ψ ₃ > equivalent up to a unitary transformation U _1→3 in {1, σ _x , σ _z , σ _x σ _z }, depending on s ₁ , s ₂ : |Ψ ₃ >=U _1→3 |Ψ ₁ > Then, steps 4 and 5 realize a wire from qubit 3 to qubit 5. The state that finally results from the procedure is |s ₁ > ₁ {circle over (×)}|s ₂ > ₂ {circle over (×)}|s ₃ > ₃ {circle over (×)}|s ₄ > ₄ {circle over (×)}|Ψ ₅ > ₅ , with |Ψ ₅ >=U _3→5 |Ψ ₃ > and U _3→5 in { ₁ , σ _x , σ _z , σ _x σ _z }. Thus, |Ψ ₅ >=U _1→5 |Ψ ₁ >, with U _l→5 =U _3→5 U _1→3 . The procedure (27) does indeed implement a wire.

[0153] Further, the composed wire from qubit 1 to 5 can be imagined as first teleporting the qubit state from qubit 1 to qubit 3 and in a second step from qubit 3 to qubit 5. This picture is very convenient, since this is in accordance with the information flow in a quantum logic network. There is, however, no temporal order of teleportations in practice. All the necessary entanglement operations are carried out in a single step during state preparation and the one-qubit measurements to implement a wire can later be performed in arbitrary temporal order.

[0154] The following describes how to deal with the randomness of the measurement results. The standard form of the gates obtained via measurement on a cluster (15) and (18) is the gate desired times some possible successive σ _x - and σ _z -rotations applied to the output qubits. The collective product operator of all these σ _x - and σ _z -operators at a certain cut through the quantum logical network will be denoted as the multi-local extra rotation U _Σ at this cut. By use of the relations (32) and (31) the unwanted extra rotations can be pulled trough the quantum logic network and summed up as one single multi-local unitary transformation applied to output state of the gate. This single local unitary transformation depends only upon the measurement outcomes obtained by imprinting the logic network on the state; it is known. Therefore, it can be corrected before the readout by a counter transformation. In other words, the bases for the readout measurements may need to be changed or, if not, at least the results of the readout measurements must be interpreted differently.

[0155] For the CNOT gate we use the following notation:

CNOT ( t _in →t _out ,c )=|0><0| _c {circle over (×)}1 ^{(t _in →t _out )} 1+|1<>1| _c {circle over (×)}σ _x ^{(t _in →t _out )} (30)

[0156] with σ _x ^{(t _in →t _out )} =|0> _{t out} <1|t _in +|1> _{t out} <0| _{t in} . A target qubit t _in is XORed with a control qubit c and thereby transferred to t _out . In the implementation discussed so far, the control qubit c remains at its site.

[0157] Now, for the CNOT and for 1-qubit rotations the following relations hold:

CNOT ( a→b,c )σ _x ^(a) =σ _x ^(b) CNOT ( a→b,c )

CNOT ( a→b,c )σ _z ^(a) =σ _z ^(b) σ _z ^(c) CNOT ( a→b,c )

CNOT ( a→b,c )σ _x ^(c) =σ _x ^(b) σ _x ^(c) CNOT ( a→b,c ) </