Fast Nonadiabatic Two-Qubit Gates for the Kane Quantum Computer

PHYSICAL REVIEW A 68, 012321 ͑2003͒

Fast nonadiabatic two-qubit gates for the Kane quantum computer

Charles D. Hill1,* and Hsi-Sheng Goan2,† 1Centre for Quantum Computer Technology, and Department of Physics, The University of Queensland, St. Lucia, Queensland 4072, Australia 2Centre for Quantum Computer Technology, University of New South Wales, Sydney, New South Wales 2052, Australia ͑Received 29 January 2003; revised manuscript received 8 May 2003; published 22 July 2003͒ In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.

DOI: 10.1103/PhysRevA.68.012321 PACS number͑s͒: 03.67.Lx

I. INTRODUCTION decomposition can be used to find optimal schemes ͓26,27͔, and of particular inspiration to this paper is an almost opti- The advent of quantum algorithms ͓1,2͔ that can outper- mal systematic method to construct the CNOT gate ͓28͔. form the best known classical algorithms has inspired many It is not possible to apply those optimal schemes ͓26,27͔ different proposals for a practical quantum computer ͓3–9͔. directly to the Kane quantum computing architecture. They One of the most promising proposals, was presented by Kane assume single-qubit gates take negligible time in comparison ͓9͔. In this proposal, a solid-state quantum computer based with two-qubit interactions, whereas on the Kane architec- on the nuclear spins of 31P atoms was suggested. Although ture, they do not. Second, in the proposal for the Kane com- initially difficult to fabricate, this scheme has several advan- puter, adjacent nuclei are coupled via the exchange and hy- tages over rival schemes ͓3–8͔. These include the compara- perfine interactions through the electrons, rather than tively long decoherence times of the 31P nuclear and electron directly, and so we have a four-‘‘qubit’’ system ͑two elec- spins ͓10–17͔, the similarity to existing Si fabrication tech- trons and two nuclei͒ rather than a two-qubit system. Al- nology, and the ability to scale. though we cannot apply optimal schemes directly, in this There have been two main proposals for pulse sequences paper we use the canonical decomposition to simplify two- to implement a CNOT ͑controlled-NOT͒ gate on the Kane qubit gate design. quantum computer. In the initial proposal ͓9͔, an adiabatic Apart from being simple to design and understand, gates CNOT gate was suggested. Since that time the details of this described in this paper have many desirable features. Some gate have been investigated and optimized ͓18–21͔. This features of these gates are the following. adiabatic scheme takes a total time of Ϸ26 ␮s and has a ͑1͒ They are simpler, with higher fidelity, and faster than systematic error of Ϸ5ϫ10Ϫ5 ͓19͔. As good as these results existing proposals. are, nonadiabatic gates have the potential to be faster with ͑2͒ They do not require sophisticated pulse shapes, such higher fidelity and allow advanced techniques such as com- as are envisioned in the adiabatic scheme, to implement. posite rotations and modified rf pulses ͓22,23͔. ͑3͒ Any two-qubit gate can be implemented directly using Wellard et al. ͓24͔ proposed a nonadiabatic pulse scheme similar schemes. This allows us to implement gates directly for the CNOT and swap gates. They present a CNOT gate that rather than as a series of CNOT gates and single-qubit rota- takes a total time of Ϸ80 ␮s with an error ͓as defined later in tions. Eq. ͑91͔͒ of Ϸ4ϫ10Ϫ4. Although this gate is nonadiabatic, This paper is organized as follows. Section II gives an it is slower than its adiabatic counterpart. For the nonadia- overview of the Kane quantum computer architecture and batic swap gate, a total time was calculated of 192 ␮s. single-qubit rotations. Section III describes the canonical de- One of the most useful tools in considering two-qubit uni- composition as it applies to the Kane quantum computer. tary interactions is the canonical decomposition ͓25–27͔. Section IV describes pulse schemes for control Z gates and This decomposition expresses any two-qubit gate as a prod- CNOT gates. Section V gives potential pulse schemes for uct of single-qubit rotations and a simple interaction content. swap and square root of swap gates. Finally, the conclusion, The interaction content can be expressed using just three Section VI, summarizes the findings of this paper. parameters. In the limit that single-qubit rotations take negligible time ͑in comparison to the speed of interaction͒, this II. THE KANE QUANTUM COMPUTER A. The Kane architecture *Electronic address: [email protected] A schematic diagram of the Kane quantum computer ar- †Mailing address: Center for Quantum Computer Technology, C/– chitecture is shown in Fig. 1. The short description given Department of Physics, The University of Queensland, St. Lucia, here follows Goan and Milburn ͓18͔. This architecture con- QLD 4072, Australia. Electronic address: [email protected] sists of 31P atoms doped in a purified 28Si (Iϭ0) host. Each

where strength A of the hyperﬁne interaction is proportional to the value of the electron wave function evaluated at the nucleus

8␲ Aϭ ␮ g ␮ ͉␺͑0͉͒2. ͑5͒ 3 B n n

A typical strength for the hyperfine interaction is Aϭ1.2 Ϫ ϫ10 4 meV. Charged A gates placed directly above each P FIG. 1. The Kane quantum computer architecture. nucleus distort the shape of the electronic wave function, thereby reducing the strength of the hyperfine coupling. The ϭ 1 P atom has nuclear spin of I 2 . Electrodes placed directly nature of this effect is under numerical investigation ͓29͔.In above each P atom are referred to as A gates and those be- this paper, we have assumed that it will be possible to vary tween atoms are referred to as J-gates. An oxide barrier sepa- the hyperfine coupling by up to Ϸ50%. rates the electrodes from the P-doped Si. The exchange interaction couples adjacent electrons. Its Each P atom has five valence electrons. As a first approxi- contribution to the Hamiltonian is mation, four of these electrons form covalent bonds with neighboring Si atoms, with the fifth forming a hydrogen-like H ϭJ␴ ␴ , ͑6͒ J e1 e2 S-orbital around each Pϩ ion. This electron is loosely bound Ϸ to the P donor and has a Bohr radius of aB* 3 nm, allowing where e1 and e2 are two adjacent electrons. The magnitude J an electron mediated interaction between neighboring nuclei. of the exchange interaction depends on the overlap of adja- In this paper, nuclear-spin states will be represented by cent electronic wave functions. J gates placed between nuclei the states ͉1͘ and ͉0͘. Electronic spin states will be repre- distort both electronic wave functions to increase or decrease sented by ͉↑͘ and ͉↓͘. Where electronic states are omitted, it the magnitude of this interaction. A typical value for the is assumed that they are polarized in the ͉↓͘ state. X, Y, and exchange energy is 4Jϭ0.124 meV, and in this paper, we Z are the Pauli matrices operating on electron and nuclear assume that it will be possible to vary the magnitude of the spins. That is, exchange interaction from Jϭ0toJϷ0.043 meV. A rotating magnetic field, of strength Bac rotating at a Xϭ␴ , Yϭ␴ , Zϭ␴ . ͑1͒ ␻ x y z frequency of ac can be applied, perpendicular to the constant magnetic field B. The contribution of the rotating mag- Operations which may be performed on any system are netic field to the Hamiltonian is governed by the Hamiltonian of the system. We now de- ϭϪ ␮ ͓ ͑␻ ͒ϩ ͑␻ ͔͒ scribe the effective spin Hamiltonian for two adjacent qubits Hac gn nBac Xn cos act Y n sin act of the Kane quantum computer and give a short physical ϩ␮ B ͓X cos͑␻ t͒ϩY sin͑␻ t͔͒, ͑7͒ motivation for each term which makes up the overall Hamil- B ac e ac e ac tonian where the strength of the rotating magnetic field is envi- Ϸ sioned to be Bac 0.0025 T. 2 At an operating temperature of Tϭ100 mK, the electrons ϭ ϩ ϩ ϩ ͑ ͒ H ͚ HB HA HJ Hac , 2 are almost all polarized by the magnetic field. That is, iϭ1 i i i ↑ n e Ϸ ϫ Ϫ12 ͑ ͒ where the summation is over each donor atom in the ↓ 2.14 10 . 8 system i. ne Under typical operating conditions, a constant magnetic ͉↓͘ field B will be applied to the entire system, perpendicular to We assume that electrons are polarized in the state, and the surface. This contributes Zeeman energies to the Hamil- use nuclear-spin states as our computational basis. tonian B. Z rotations ϭϪ ␮ ϩ␮ ͑ ͒ HB gn nBZn BBZe . 3 Single-qubit rotations are required to implement the two- qubit gates described in this paper, as well as being essential A typical value for the Kane quantum computer of B for universality. In fact, as we will see they contribute sig- ϭ ␮ nificantly to the overall time and fidelity of each two-qubit 2.0 T gives Zeeman energy for the electrons of BB Ϸ ␮ Ϸ ϫ Ϫ5 gate. It is therefore important to consider the time required to 0.116 meV and for the nucleus gn nB 7.1 10 meV. The hyperfine interaction couples between nuclear and implement Z, X, and Y rotations. electronic spins. The contribution of the hyperfine interaction In this section we describe how fast Z rotations may be to the Hamiltonian is performed varying the voltage on the A gates only. A Z rotation is described by the equation ϭ ␴ ␴ ͑ ͒ ͑␪͒ϭ i␪Z/2 ͑ ͒ HA A e n , 4 Rz e . 9

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TABLE I. Typical parameters for a Z rotation. TABLE II. Typical parameters for an X rotation.

Description Term Value ͑meV͒ Description Term Value

Unperturbed hyperfine interaction A 0.1211ϫ10Ϫ3 Unperturbed hyperfine interaction A 0.1211ϫ10Ϫ3 meV ϫ Ϫ3 ϫ Ϫ3 Hyperfine interaction during Z rotation Az 0.0606 10 Hyperfine interaction during X rotation Ax 0.0606 10 meV Constant magnetic-field strength B 2.000 T

Rotating magnetic-field strength Bac 0.0025 T A Z gate ͑phase flip͒ may be implemented as a rotation. It is given up to a global phase by X and Y rotations are performed by application of a rotating ϭϪ ͑␲͒ ͑ ͒ Z iRz . 10 magnetic field Bac . The rotating magnetic field is resonant with the Larmor precession frequency given in Eq. ͑11͒, that Under the influence of a constant magnetic field B to sec- is, ond order in A ͓18͔, each nuclei will undergo Larmor preces- ␻ ϭ␻ ͑ ͒ sion around the Z axis, at frequency of ac l . 17

2A2 In contrast to NMR, in the Kane proposal we have direct ប␻ ϭ2g ␮ Bϩ2Aϩ . ͑11͒ l n n ␮ Bϩg ␮ B control over the Larmor frequency of each individual P B n n nucleus. By reducing the hyperfine coupling for the atom, we wish to target from A to Ax , we may apply an oscillating Z rotations may be performed by variation of the hyperfine magnetic field that is only resonant with the Larmor fre- interaction from A to Az giving a difference in rotation frequency of only one of the atoms. This allows us to induce an quency of X or Y rotation on an individual atom. To the first order, the frequency of this rotation may be approximated by ͑ 2Ϫ 2͒ 2 A Az ប␻ ϭ2͑AϪA ͒ϩ . ͑12͒ z z ␮ ϩ ␮ A BB gn nB ប␻ ϭ ␮ ͩ ϩ x ͪ ͑ ͒ x gn nBac 1 ␮ . 18 gn nB Perturbing the hyperfine interaction for one of the atoms and allowing free evolution, will rotate this atom with re- The speed of an X rotation is directly proportional to the spect to the rotation of the unperturbed atoms. The speed of strength of the rotating magnetic field Bac . As the strength single atom Z rotations depends on how much it is possible of the rotating magnetic field Bac increases, the fidelity of the to vary the strength of the hyperfine interaction A. For nu- operation decreases. The reason is that in frequency space merical simulation we use the typical values shown in the full width at half maximum of the transition excited by Table I. the rotating magnetic field increases in proportion to Bac . Under these conditions, a Z gate may be performed on a That is, as Bac increases, we begin to excite nonresonant single nuclear spin in approximately transitions. The larger separation, in frequency space, between Larmor frequencies, the smaller this systematic error. Ϸ ␮ ͑ ͒ tZ 0.021 s. 13 Since the Larmor precession frequency depends on how much we are able to vary the hyperfine interaction A,itde- These rotations occur in a rotating frame that precesses termines how strong we are able to make Bac. around the Z axis with a frequency equal to the Larmor fre- For the purpose of simulation, the typical values shown in quency. We may have to allow a small time of free evolution Table II for the unperturbed hyperfine interaction strength A, until nuclei that are not affected by the Z rotation orient the hyperfine interaction strength during the X rotation Ax , themselves to their original phase. The time required for this applied magnetic-field strength B, and rotating magnetic- operation is less than field strength Bac were used. Using these parameters, this gives the overall time to per- р ␮ ͑ ͒ form an X gate on a single-qubit in approximately tF 0.02 s. 14 Ϸ ␮ ͑ ͒ tX 6.4 s. 19 C. X and Y rotations In this section, we show how techniques, similar to those Any single-qubit gate may be expressed as a product of X, used in NMR ͑nuclear magnetic resonance͓͒18,30,31͔, may Y and Z rotations. Ideally, X and Y rotations should be mini- be used to implement X and Y rotations. X and Y rotations are mized because Z rotations may be performed much faster described by the equations than X or Y rotations. For example, a Hadamard gate may be expressed as a product of Z and X rotations: ␪͒ϭ i␪X/2 ͑ ͒ Rx͑ e , 15 ␲ ␲ ␲ ϭ ͩ ͪ ͩ ͪ ͩ ͪ ͑ ͒ ␪͒ϭ i␪Y/2 ͑ ͒ H Rz Rx Rz . 20 Ry͑ e . 16 2 2 2

012321-3 C. D. HILL AND H.-S. GOAN PHYSICAL REVIEW A 68, 012321 ͑2003͒

Thus, from the above discussion, the Hadamard gate takes a TABLE III. Typical parameters during interaction. time of approximately Description Term Value ͑meV͒ t Ϸ3.2 ␮s. ͑21͒ H ϫ Ϫ3 Hyperﬁne interaction during interaction AU 0.1197 10 Exchange interaction during interaction J 0.0423 D. Nuclear-spin interaction U In this section, we show the results of second order perturbation theory to describe the interaction between two Since we are free to choose our zero-point energy to be E0 neighboring P atoms. This interaction between nuclei is ͑or equivalently ignore a global phase of a wave function coupled by electron interactions. We consider the case where ͉␺͘) we may rewrite the second-order approximation as the hyperﬁne couplings between each nucleus and its elec- ϭប␻ ͑ ͒ tron are equal, that is, E͉↓↓͉͘11͘ B , 32

AϭA ϭA . ͑22͒ E͉↓↓͉͘ ͘ϭប␻ , ͑33͒ 1 2 sn S We allow coupling between electrons, that is, E͉↓↓͉͘ ͘ϭϪប␻ , ͑34͒ an S JϾ0, ͑23͒ ϭϪប␻ ͑ ͒ E͉↓↓͉͘00͘ B , 35 but restrict ourselves to be far from an electronic energy- ␻ ␻ level crossing where B and S are given by ␮ 2 2 BB A A JӶ . ͑24͒ ប␻ ϭ ϩ ␮ ϩ ϩ B 2A 2gn nB ␮ ϩ ␮ ␮ ϩ ␮ Ϫ , 2 BB gn nB BB gn nB 2J Under these conditions electrons will remain in the polarized ͑36͒ ͉↓↓͘ ground state. A2 A2 In this situation, analysis has been performed using the ប␻ ϭ Ϫ ͑ ͒ ͓ ͔ S ␮ ϩ ␮ Ϫ ␮ ϩ ␮ . 37 second-order perturbation theory 18 . To second order in A, BB gn nB 2J BB gn nB the energy levels are The reason for this representation of the energy will become ϭϪ ␮ ϩ ϩ ␮ ϩ ͑ ͒ E͉11͘ 2 BB J 2gn nB 2A, 25 clear in the following section. Typical values used during numerical simulation of the interaction between nuclei are 2 2A shown in Table III. E͉ ͘ϭϪ2␮ BϩJϪ , ͑26͒ sn B ␮ ϩ ␮ BB gn nB III. THE CANONICAL DECOMPOSITION 2A2 E͉ ͘ϭϪ2␮ BϩJϪ , ͑27͒ an B ␮ Bϩg ␮ BϪ2J In this section, we describe the canonical decomposition B n n and describe how this decomposition may be applied to the 2A2 Kane quantum computer. ϭϪ ␮ ϩ Ϫ ␮ Ϫ Ϫ E͉00͘ 2 BB J 2gn nB 2A ␮ ϩ ␮ Ϫ BB gn nB 2J A. Mathematical description of canonical decomposition 2A2 Ϫ ͑ ͒ The canonical decomposition ͓25,27͔ decomposes any ␮ ϩ ␮ , 28 BB gn nB two-qubit unitary operator into a product of four single-qubit unitaries and one entangling unitary: ͉ ͉ where the symmetric sn͘ and antisymmetric an͘ energy ϭ ͒ ͒ ͑ ͒ eigenstates are given by U ͑V1 V2 Ucan͑W1 W2 , 38

1 where V , V , W , and W are single-qubit unitaries, and ͉ ϭ ͉ ϩ͉ ͑ ͒ 1 2 1 2 sn͘ ͑ 10͘ 01͘), 29 ͱ2 Ucan is the two-qubit interaction. The symbol represents the tensor product of two matrices. U has a simple form involving only three parameters, 1 can ͉ ϭ ͉ Ϫ͉ ͑ ͒ ␣ , ␣ , and ␣ : an͘ ͑ 10͘ 01͘). 30 x y z ͱ2 ␣ ␣ ␣ ϭ i xX X i yY Y i zZ Z ͑ ͒ Ucan e e e . 39 Notice that the energies are symmetric around This purely nonlocal term is known as the interaction con- 2 2 A A tent of the gate. It is not difﬁcult to show that each of the E ϭϪ2␮ BϩJϪ Ϫ . ␣ ␣ 0 B ␮ ϩ ␮ Ϫ ␮ ϩ ␮ i xX X i yY Y BB gn nB 2J BB gn nB terms in the interaction content, e , e , and ␣ ͑31͒ ei zZ Z, commute with each other.

012321-4 FAST NONADIABATIC TWO-QUBIT GATES FOR THE... PHYSICAL REVIEW A 68, 012321 ͑2003͒

␣ ␣ Physically each of the terms ei xX X, ei yY Y, and which we may decompose using the canonical decomposi- ␣ ei zZ Z correspond to a type of controlled rotation. For ex- tion: ample, following Ref. ͓28͔, ϭ͑ s s ͒ s ͑ s s ͒ ͑ ͒ Usys V1 V2 Ucan W1 W2 , 45 ␣ ei zZ Zϭcos ␣ I Iϩi sin ␣ Z Z z z where the superscript s indicates a physical operation present ϭ ␣ ͉ ͉ϩ͉ ͉͒ cos z͑ 0͗͘0 1͗͘1 I in our system. We wish to find the interaction content Us of this free ϩ ␣ ͉͑ ͗͘ ͉Ϫ͉ ͗͘ ͉͒ can i sin z 0 0 1 1 Z evolution. Systematic methods for doing this are given in ␣ Ϫ ␣ ͓ ͔ ϭ͉0͗͘0͉ ei zZϩ͉1͗͘1͉ e i zZ Refs. 25,26,33 . This is most easily done by noting any interaction content, Ucan is diagonal in the so-called magic ␣ Ϫ ␣ ϭ͑I ei zZ͉͒͑0͗͘0͉ Iϩ͉1͗͘1͉ e i2 zZ͒. basis, otherwise known as the Bell basis. This basis is ͑ ͒ 40 1 ͉⌽ ͘ϭ ͉͑ ͘ϩ͉ ͘ ͑ ͒ ␣ 1 00 11 ), 46 This shows that up to a single-qubit rotation, ei zZ Z is ͱ2 equivalent to a controlled Z rotation. This holds true for the other two terms. If we denote the eigenstates of X by Ϫi ͉⌽ ϭ ͉ Ϫ͉ ͑ ͒ 2͘ ͑ 00͘ 11͘), 47 ͱ2 X͉xϩ͘ϭϩ͉xϩ͘, ͑41͒ 1 X͉xϪ͘ϭϪ͉xϪ͘, ͑42͒ ͉⌽ ϭ ͉ Ϫ͉ ͑ ͒ 3͘ ͑ 01͘ 10͘), 48 ͱ2 then a similar analysis shows that Ϫi Ϫi␣ X i␣ X X Ϫi2␣ X ͉⌽ ͘ϭ ͉͑ ͘ϩ͉ ͘ ͑ ͒ ͑I e x ͒e x ϭ͉xϩ͗͘xϩ͉ Iϩ͉xϪ͗͘xϪ͉ e x , 4 01 10 ). 49 ͱ2 ͑43͒ ␭ ␭ ␣ ␣ ␣ i 1 i 2 x , y , and z are related to the eigenvalues e , e , ␭ ␭ i 3 i 4 and that e , and e of Ucan . That is, ␭ ϭϩ␣ Ϫ␣ ϩ␣ ͑ ͒ Ϫi␣ Y i␣ Y Y Ϫi2␣ Y 1 x y z , 50 ͑I e y ͒e y ϭ͉y ϩ͗͘y ϩ͉ Iϩ͉y Ϫ͗͘y Ϫ͉ e y . ␭ ϭϪ␣ ϩ␣ ϩ␣ ͑ ͒ ͑44͒ 2 x y z , 51 ␭ ϭϪ␣ Ϫ␣ Ϫ␣ , ͑52͒ These operations are equivalent to controlled rotations in the 3 x y z X and Y directions, respectively. For the first case, if the ␭ ϭϩ␣ ϩ␣ ϩ␣ ͑ ͒ 4 x y z . 53 control qubit is in the ͉xϪ͘ state an X rotation is applied to the target qubit, and not applied if the control qubit is in the It is possible to relate these eigenvalues to our system. ͉xϩ͘ state. Similarly for Y. After a time t, each of the eigenstates of the system will have ¨ Single-qubit rotations, V1 , V2 , W1 , W2, are possible on evolved according the Schrodinger equation, which we may the Kane quantum computing architecture, the remaining view as having performed an operation Usys(t) on the sys- task is to specify the pulse sequence for the purely entan- tem. As we showed in Sec. II D gling unitary U . Fortunately, this is always possible, as ϩ ␪ can U ͉11͘ϭe i B͉11͘, ͑54͒ any interaction ͑with single-qubit rotations͒ between the two sys Ϫ ␪ nuclei is sufficient ͓32͔. In fact, it is a relatively simple task ͉ ϭ i B͉ ͑ ͒ Usys 00͘ e 00͘, 55 to use almost any interaction between qubits to generate any ϩ ␪ ͉ ϭ i S͉ ͑ ͒ desired operation. Usys s͘ e s͘, 56 Ϫ ␪ U ͉a͘ϭe i S͉a͘, ͑57͒ B. Calculation of the interaction content between nuclei sys In this section, we will see how it is possible to apply the where canonical decomposition to the Kane quantum computer. ␪ ϭ␻ t, ͑58͒ This is important as this natural interaction of the system will S S be manipulated by single-qubit unitaries to find the pulse ␪ ϭ␻ ͑ ͒ B Bt. 59 scheme of any two-qubit gate. The canonical decomposition provides a unique way of looking at this interaction. Applying Eqs. ͑54͒–͑57͒ to Eqs. ͑46͒–͑49͒, we obtain The interaction that we will apply the canonical decom- U ͉⌽ ͘ϭcos͑␪ ͉͒⌽ ͘Ϫsin͑␪ ͉͒⌽ ͘, ͑60͒ position to is free evolution of the configuration described in sys 1 B 1 B 2 Sec. II D, using the results cited there from the second-order ͉⌽ ϭ ␪ ͉͒⌽ ϩ ␪ ͉͒⌽ ͑ ͒ Usys 2͘ cos͑ B 2͘ sin͑ B 1͘, 61 perturbation theory. After a particular time of free evolution, Ϫ ␪ ͉⌽ ϭ i S͉⌽ ͑ ͒ our system will have evolved according to unitary dynamics, Usys 3͘ e 3͘, 62

012321-5 C. D. HILL AND H.-S. GOAN PHYSICAL REVIEW A 68, 012321 ͑2003͒

ϩ ␪ ͉⌽ ϭ i S͉⌽ ͑ ͒ Usys 4͘ e 4͘. 63 This analysis has been based on the second order perturbation theory. As we approach the electronic energy-level This shows that in the magic basis, Usys is given by crossing, this approach is no longer valid. Close to this crossing numerical analysis shows the eigenvalues are no longer ͑␪ ͒ ͑␪ ͒ cos B sin B 00 ␣s symmetric which implies z becomes nonzero. Unfortu- Ϫ ␪ ͒ ␪ ͒ sin͑ B cos͑ B 00 nately, in this regime, we excite the system into higher- ϭ Ϫi␪ ͑ ͒ energy electronic conﬁgurations. Usys ͫ 00e S 0 ͬ . 64 Given any two-qubit gate, such as the CNOT gate, there are i␪ 000e S many different possible choices of single-qubit rotations and free evolution that will implement a desired gate. Z rotations ␭ ␭ ␭ ␭ are faster single-qubit rotations than X and Y rotations, and It is possible to ﬁnd the eigenvalues 1 , 2 , 3, and 4. We note that the eigenvalues of UTU in the magic basis are therefore, it is desirable to minimize X and Y rotations in given by order to optimize the time required, for any given two-qubit gate. ␭ ␭ ␭ ␭ ␭͑UTU͒ϭ͕e2i 1,e2i 2,e2i 3,e2i 4͖. ͑65͒

T IV. THE CNOT AND CONTROLLED Z GATES Calculation of the eigenvalues of UsysUsys is easy since T UsysUsys is already diagonal in this basis, with diagonal el- A. Introduction Ϫ ␪ ␪ ements being ͕1,1,e 2i S,e2i S͖. Care must be exercised at The CNOT gate is a particularly often cited example of a this point because it is not clear which branch should be used р␪ two-qubit gate. CNOT and single-qubit rotations are universal when taking the argument. In our case, as long as 0 S ͓ ͔ р ␲ ͓ ͔ for quantum computation 34 . Many implementations, in- ( /2) 33 , then cluding the Kane proposal ͓9͔, use this fact to demonstrate ␭ ϭ0, ͑66͒ that they can, in principle, perform any quantum algorithm. 1 It is a member of the so-called fault tolerant ͓35͔ set of gates, ␭ ϭ0, ͑67͒ which are universal for quantum computing, and are particu- 2 larly important in error correction. In this section, we find a ␭ ϭϪ␪ ͑ ͒ pulse scheme to implement the CNOT gate on the Kane quan- 3 S , 68 tum computer. ␭ ϭϩ␪ ͑ ͒ Controlled Z rotations, sometimes known as controlled 4 S . 69 phase gates, are some of the most important operations for ͑ ͒ ͑ ͒ ␣ Using Eqs. 50 – 53 we may solve for the coefficients x , implementing quantum algorithms. In particular, one of the ␣ ␣ y , and z , giving simplest ways to implement quantum Fourier transforma- tions uses multiple controlled Z rotations ͑see, for example, 1 Ref. ͓36͔͒. Single-qubit rotations and the controlled Z gate ␣s ϭ ␪ , ͑70͒ x 2 S are, like the CNOT gate, universal for quantum computation. Controlled Z rotations may be used in the construction of 1 controlled X and Y rotations. In this section we find a pulse ␣s ϭ ␪ , ͑71͒ y 2 S scheme to implement any controlled Z rotation on the Kane quantum computer. ␣sϭ0. ͑72͒ Because these two gates have similar interaction contents, z we consider them together. We will first show how to con- s s s s struct a controlled-Z gate of any angle and use this gate di- Single-qubit rotations, W1 , W2 , V1 , V2 , induced are Z rotations. Z rotations are fast and may be canceled in com- rectly to construct a CNOT gate. ␪ paratively less time by single-qubit Z rotations in the oppo- A controlled Z rotation of angle is defined in the com- site direction: putational basis by

͑ s† s†͒ ͑ s† s†͒ϭ s ͑ ͒ 100 0 V1 V2 Usys W1 W2 Ucan . 73 010 0 ͑␪͒ϭ ͑ ͒ For notational convenience, we will now label the interaction U⌳Z ͫ ͬ . 76 001 0 content of the system by an angle rather than by its time. The ␪ time for this interaction may be calculated through Eqs. 000ei ͑70͒–͑72͒, ͑58͒, and ͑37͒. Therefore, we write The canonical decomposition of the controlled Z rotation by s ͑␾͒ϭ i␾X Xϩi␾Y Y ͑ ͒ Ucan e , 74 an angle ␪ has an interaction content consisting of where ␣ ϭ ͑ ͒ x 0, 77 1 ␾ϭ ␪ ͑ ͒ S . 75 ␣ ϭ ͑ ͒ 2 y 0, 78

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␪ ␣ ϭ . ͑79͒ z 2

This interaction content may be found by using systematic methods ͓25,26,33͔. The controlled-Z gate also requires a Z rotation as described by Eq. ͑40͒. CNOT is deﬁned in the computational basis by the matrix FIG. 2. Circuit diagram for controlled Z pulse sequence.

1000 ␣ ϩ ␣ ϩ ␣ ͑H H͒ei xX X i yY Y i zZ Z͑H H͒

0100 ␣ ϩ ␣ ϩ ␣ ϭ ͑ ͒ ϭ i zX X i yY Y i xZ Z ͑ ͒ UCNOT ͫ ͬ . 80 e . 86 0001 0010 Combining these two techniques gives the following construction: The canonical decomposition of CNOT has an interaction ␪ content with angles of ei␪Z Zϭ͑Z I͒͑H H͒Us ͩ ͪ ͑H H͒͑Z I͒ can 2 ␣ ϭ ͑ ͒ x 0, 81 ␪ ϫ͑ ͒ s ͩ ͪ ͑ ͒ ͑ ͒ H H Ucan H H . 87 ␣ ϭ ͑ ͒ 2 y 0, 82 To find the final construction, several one-qubit optimiza- ␲ tions were made by combining adjacent single-qubit rota- ␣ ϭ . ͑83͒ z 4 tions and using the identities ϭ ͑ ͒ Since the CNOT and controlled-Z gates are both types of HZH X, 88 controlled rotation similar to those described in Sec. III A, it ϭ ͑ ͒ is not a surprise that they have a similar interaction content. HH I. 89 In fact, control-Z gates ͑that is, a controlled Z rotation by an Operations may be performed in parallel. For example, angle of ␲) and CNOT gates have an identical interaction content, and are therefore equivalent up to single-qubit rota- performing identical X or Y rotations on separate nuclei is a natural operation of the system because magnetic fields are tions. A CNOT gate may be constructed from a control-Z gate conjugated by I H. applied globally. Performing operations in parallel is faster and also have higher fidelity than performing them one at a time. B. The construction The construction of the controlled Z rotation is shown in Our first task in finding a suitable pulse scheme for the Fig. 2. In this circuit, the single-qubit rotations specified in controlled Z rotation is to find a pulse scheme which imple- Eq. ͑40͒ have been included. The period of interaction be- ments the interaction content ͓Eqs. ͑77͒–͑79͔͒ of the con- tween nuclei may be increased or decreased to produce controlled Z rotation. Techniques have direct analogs in NMR trolled rotations by any angle ␪ as specified in Eqs. ͑87͒, ͓30,31͔. ͑74͒, ͑58͒, and ͑37͒. The first technique ͓28͔ is to conjugate by I X, I Y,or Our task of constructing a CNOT gate is now compara- I Z to change the sign of two of these parameters. For tively simple. We note that a CNOT gate has the same inter- example, action term as the controlled-Z ͑controlled phase͒ operation. These gates are therefore equivalent up to local operations. i␣ X Xϩi␣ Y Yϩi␣ Z Z ͑I Z͒e x y z ͑I Z͒ Conjugation by I H will turn a controlled-Z operation

Ϫ ␣ Ϫ ␣ ϩ ␣ into a CNOT gate. Using some simple one qubit identities to ϭe i xX X i yY Y i zZ Z. ͑84͒ simplify the rotations at the beginning and end of the pulse sequences, we arrive at the decomposition illustrated in the This can be useful because it allows us to exactly cancel circuit diagram shown in Fig. 3. every controlled rotation except one

␣ ͒ ͒ ϭ i2 zZ Z ͑ ͒ ͑I Z Ucan͑I Z Ucan e . 85

␣sϭ In our case, however, it turns out that z 0. In order to reorder the parameters, a useful technique is to conjugate by Hadamardgates ͓28͔. This is one of only several choices of single-qubit rotations which reorder the parameters. In this case, the order of the parameters is FIG. 3. Circuit diagram for the CNOT pulse sequence.

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FIG. 4. Numerical simulation of the CNOT gate showing different initial conditions.

C. Time and fidelity gate, only 3.2 ␮s is spent implementing the entangling part Throughout the paper, we define fidelity as of the gate, whereas 12.6 ␮s is required to implement the X and Y rotations. ͉␺ ͉␺ ϭ͉ ␺͉␺ ͉2 ͑ ͒ F͑ ͘, 0͘) ͗ 0͘ , 90 We can see via simulation that the systematic error in the Ϫ CNOT gate is Ϸ4ϫ10 5. Some of these error will be due to errors during simulation and breakdown of the second-order with ͉␺͘ being the actual state obtained from evolution and approximation. A large part of the error, particularly if the ͉␺ ͘ being the state which is desired. We define the error in 0 hyperfine interaction may not be varied very much, is due to terms of the fidelity as X rotations where unintended nonresonant transitions are excited along with the intended rotation. ϭ Ϫ ͉␺ ͉␺ ͑ ͒ E max͓1 F͑ ͘, 0͘)], 91 ͉␺͘ V. THE SWAP AND SQUARE ROOT OF SWAP GATES A. Introduction where the maximization is performed over the output of all the computational basis states ͉␺͘. One of the most important gates for the Kane quantum Numerical simulations were carried out by numerically computer is envisioned to be the swap gate. This is because, integrating Schro¨dinger’s equation for the Hamiltonian of the in the Kane proposal, only nearest-neighbor interactions are system, Eq. ͑2͒. The results of this numerical simulation for allowed. This gate swaps the quantum state of two-qubits. By using the swap gate it is possible to swap qubits until the pulse sequence of the CNOT gate are shown in Fig. 4. These graphs show each of the states and the transitions they are the nearest neighbors, interact with them, and then which are made. In these figures, it is possible to see the swap them back again. Having an efficient method to interact evolution of each of the four computational basis states. The qubits which are not adjacent to each other is therefore im- control qubit is the second qubit and the target qubit is the first qubit. TABLE IV. Time for the CNOT gate. According to the numerical results, a full controlled-Z Description Time ␮s gate takes a total time of 16.1 ␮s and has an error of Ϸ4 Ϫ ϫ10 5. Similarly, we find the CNOT gate takes a total time X rotations 12.6 of 16.0 ␮s. The time required for this gate can be grouped as Z rotations 0.2 shown in Table IV. Two qubit interaction 3.2 X and Y rotations make up the majority of the time taken Total 16.0 to implement the controlled-Z and CNOT gates. In the CNOT

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FIG. 6. The circuit diagram for the square root of the swap pulse FIG. 5. Circuit diagram for the swap gate pulse sequence. sequence.

portant, and the swap gate, with its high level of information ␲ ␣ ϭ , ͑94͒ transfer, is one possible method of achieving this. y 4 The square root of swap gate has been suggested for the quantum-dot-spin based quantum computer architecture ͓37͔, ␲ where it is a particularly natural operation. In our system, it ␣ ϭ . ͑95͒ is not such a natural operation, but that does not mean that z 4 we cannot construct it. Like the CNOT gate, the square root of ͑ ͒ swap together with single-qubit rotations is universal for The square root of the swap gate is defined in the compu- quantum computation. In this section, we find a pulse se- tational basis by quence to implement both the swap and the square root of swap gates on the Kane quantum computer architecture. The swap gate is defined in the computational basis by 10 00 1 1 1000 0 ͑1ϩi͒ ͑1Ϫi͒ 0 2 2 ϭ ͑ ͒ 0010 USS . 96 ϭ ͑ ͒ 1 1 Uswap ͫ ͬ . 92 0 ͑1Ϫi͒ ͑1ϩi͒ 0 0100 ΄ ΅ 2 2 0001 00 01

The canonical decomposition of the swap gate has an inter- The canonical decomposition of the square root of the swap action content with angles of gate has an interaction term consisting of

␲ ␲ ␣ ϭ , ͑93͒ ␣ ϭϪ , ͑97͒ x 4 x 8

FIG. 7. Numerical simulation of the swap gate.

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TABLE V. Gate times and fidelities. three adiabatic CNOT gates, which would take Ϸ78 ␮s. According to numerical simulation the square root of ␮ Gate Time ( s) Error swap gate takes 16.8 ␮s and has an error of approximately Ϫ Ϫ ϫ 5 CNOT 16.0 4ϫ10 5 5 10 . This is the first explicit proposal for the Kane Swap 19.2 7ϫ10Ϫ5 quantum computer for the square root of swap gate. Square root of swap 16.2 5ϫ10Ϫ5 The square root of swap gate has been suggested in the Ϫ ͓ ͔ Controlled Z 16.1 4ϫ10 5 context of quantum computation for quantum dots 37 .Itis universal for quantum computation and therefore can be used to construct a CNOT gate. Unfortunately in this case, a CNOT ␲ constructed from the square root of swap gate presented here ␣ ϭϪ , ͑98͒ ␮ y 8 would take approximately 40 s which is much longer than the pulse sequence presented in this paper for the CNOT gate. ␲ ␣ ϭϪ . ͑99͒ z 8 VI. CONCLUSION We have shown how the canonical decomposition may be Since the square root of swap and swap gates have essen- applied to the Kane quantum computer. We found the ca- tially the same interaction content, their constructions are nonical decomposition of a natural operation of the com- very similar, and are therefore considered together here. puter, that is, free evolution with hyperfine interactions equal and the exchange interaction non-zero. We then used this B. The construction interaction to form two-qubit gates which may be applied to The easiest way to construct a swap gate is simply to use the Kane quantum computer. These gates and their times and ␣ ␣ free evolution to obtain the angles x and y which is natural fidelities are shown in Table V. ␣ for our system. The only remaining term is the z term, The majority of the time required to implement each of which for our system will naturally be zero. We may obtain these two-qubit gates is used to implement single-qubit rota- this term by applying a pulse sequence similar to the con- tions. Were we able to perform these rotations faster and trolled Z rotation as described in Sec. IV. The resulting con- more accurately then the gates presented here would also struction swap gate is shown in the diagram in Fig. 5. benefit. Another possible avenue of research is to investigate The interaction content of the square root of swap gate is the effect of decoherence on the system. exactly half that of the swap gate, and it is negative. We use To our knowledge, this is the fastest proposal for swap, exactly the same technique used to obtain the swap gate, square root of swap, CNOT, and controlled-Z operations on only allowing the nuclei to interact for exactly half the time. the Kane quantum computer architecture. We have shown To make the terms negative, we conjugate by Z I. The how a representative set of two-qubit gates may be imple- construction of the square root of swap gate obtained using mented on the Kane quantum computer. These methods may this method is shown in Fig. 6. prove particularly powerful because they only involve char- acterization by three parameters which may be determined C. Speed and fidelity theoretically, as shown here, or through experiment. Once determined, these parameters may be used to construct any The swap and square root of swap gates were simulated two-qubit gate. numerically. The resulting transitions for the swap gate are shown in Fig. 7. Similar results were obtained for the square root of swap gate, not shown here. ACKNOWLEDGMENTS The swap gate takes a total time of 19.2 ␮s, and has a fidelity of Ϸ7ϫ10Ϫ5. The majority of time in this gate is We would like to thank Gerard Milburn for support. taken by X and Y rotations, which are also the major sources C.D.H. would like to thank Mick Bremner, Jennifer Dodd, of error. Henry Haselgrove, and Tobias Osborne for help and advice. This is substantially faster than an existing suggestion for H.S.G. would like to acknowledge financial support from the swap gate ͓19͔ of 192 ␮s. It is also faster than using Hewlett-Packard.

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