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Universal Quantum Computation Using Exchange Interactions and Measurements of Single- and Two-Spin Observables

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Universal Quantum Computation Using Exchange Interactions and Measurements of Single- and Two-Spin Observables RAPID COMMUNICATIONS PHYSICAL REVIEW A 67, 050303͑R͒͑2003͒ Universal quantum computation using exchange interactions and measurements of single- and two-spin observables Lian-Ao Wu and Daniel A. Lidar Chemical Physics Theory Group, Chemistry Department, University of Toronto, 80 St. George Street, Toronto, Ontario M5S 3H6, Canada ͑Received 17 August 2002; published 28 May 2003͒ We show how to construct a universal set of quantum logic gates using control over exchange interactions and single- and two-spin measurements only. Single-spin unitary operations are teleported between neighbor- ing spins instead of being executed directly, thus potentially eliminating a major difficulty in the construction of several of the most promising proposals for solid-state quantum computation, such as spin-coupled quantum dots, donor-atom nuclear spins in silicon, and electrons on helium. Contrary to previous proposals dealing with this difficulty, our scheme requires no encoding redundancy. We also discuss an application to superconducting phase qubits. DOI: 10.1103/PhysRevA.67.050303 PACS number͑s͒: 03.67.Lx, 05.30.Ch, 03.65.Fd I. INTRODUCTION the linear optics case, it was shown that photon-photon in- teractions can be induced indirectly via gate teleportation Quantum computers ͑QCs͒ hold great promise for inher- ͓6͔. This idea has its origins in earlier work on fault-tolerant ently faster computation than is possible on their classical constructions for quantum gates ͓7͔ and stochastic program- counterparts, but so far progress in building a large-scale QC mable quantum gates ͓8͔. The same work inspired more re- has been slow. An essential requirement is that a QC should cent results showing that, in fact, measurements and state be capable of performing ‘‘universal quantum computation’’ preparation alone suffice for UQC ͓9–11͔. ͑UQC͒: a set of quantum logic gates ͑unitary transforma- Experimentally, retaining only the absolutely essential in- tions͒ is said to be ‘‘universal’’ if any unitary transformation gredients needed to construct a universal QC may be an im- can be approximated to arbitrary accuracy by a quantum cir- portant simplification. Since read-out is necessary, measure- cuit involving only those gates ͓1͔. One of the chief obstacles ments are inevitable. Here, we propose a minimalistic in constructing large-scale QCs is the seemingly innocuous, approach for universal quantum computation that is particu- but in reality very daunting set of requirements must be met larly well suited to the important class of spin-based QC for universality, according to the standard circuit model ͓1͔: proposals governed by exchange interactions ͓2–4͔, and ͑1͒ preparation of a fiducial initial state ͑initialization͒, ͑2͒ a other proposals governed by effective exchange interactions set of single and two-qubit unitary transformations generat- ͓12͔. In particular, we show that UQC can be performed ing the group of all unitary transformations on the Hilbert using only single- and two-qubit measurements and con- space of the QC ͑computation͒, and ͑3͒ single-qubit mea- trolled exchange interactions, via gate teleportation. We has- surements ͑read out͒. Since initialization can often be per- ten to add that in our case teleportation involves only formed through measurements, requirements ͑1͒ and ͑3͒ do nearest-and next-nearest-neighbor spins, so that no coherent not necessarily imply different experimental procedures and manipulations between macroscopically separated spins are contraints. Until recently it was thought that computation is required. In our approach, which offers a new perspective on irreducible to measurements, so that requirement ͑2͒ would the requirements for UQC, two important advantages are ob- appear to be an essential component of UQC. However, uni- tained: ͑i͒ we require no encoding redundancy, ͑2͒ the need tary transformations are sometimes very challenging to per- to perform the aforementioned difficult single-spin unitary form. Two important examples are the exceedingly small operations is obviated, and replaced by measurements, which photon-photon interaction that was thought to preclude linear are anyhow necessary. The tradeoff is that the implementa- optics QCs, and the difficult-to-execute single-spin gates in tion of gates becomes probabilistic ͑as in all gate- certain solid-state QC proposals, such as quantum dots ͓2͔ teleportation-based approaches͒, but this probability can be and donor atom nuclear or electron spins in silicon ͓3,4͔. The boosted arbitrarily close to 1, exponentially fast in the num- problem with single-spin unitary gates is that they impose ber of measurements. difficult demands on g-factor engineering of heterostructure materials, and require strong and inhomogeneous magnetic II. SUPERCONDUCTING PHASE QUBITS EXAMPLE fields or microwave manipulations of spins, that are often slow and may cause device heating ͓5͔. For this reason there We begin our discussion with a relatively simple example has recently been a great deal of theoretical activity involv- of the utility of measurement-aided UQC, in the context of ing various qubit encoding schemes, which allow for UQC the proposal to use d-wave grain boundary ͑dGB͒ phase qu- without invoking difficult-to-control single-spin gates. Spe- bits for QC ͓13͔. In this proposal, it is important to reduce cifically, in the case of exchange Hamiltonians, it was shown the constraints on fabrication by removing the need to apply that when qubits are encoded into states of two or more a bias on individual qubits ͓14͔. This bias requires, e.g., the spins, the exchange interaction, possibly supplemented by possibility of applying a local magnetic field on each qubit, static Zeeman splittings, is sufficient to construct a set of and is a major experimental challenge. The effective system ϭ ϩ universal gates; see, e.g., Ref. ͓5͔ and references therein. In Hamiltonian that we consider is then HS HX HZZ , where 1050-2947/2003/67͑5͒/050303͑4͒/$20.0067 050303-1 ©2003 The American Physical Society RAPID COMMUNICATIONS L.-A. WU AND D. A. LIDAR PHYSICAL REVIEW A 67, 050303͑R͒͑2003͒ ϭ͚ ⌬ HX i iXi describes phase tunneling, and HZZ ϭ͚ i, jJijZiZ j represents Josephson coupling of qubits; ␴x ␴y ␴z Xi ,Y i ,Zi denote the Pauli matrices , , acting on the ith qubit. We assume continuous control over Jij . In Ref. ͓14͔ it was shown how UQC can be performed given this Hamiltonian, by encoding a logical qubit into two physical qubits, and using sequences of recoupling pulses. Instead, we now show how to implement Zi using measurements, which together with HS is sufficient for UQC. Suppose we start from an unknown state of qubit 1: ͉␺͘ϭa͉0͘ϩb͉1͘.By ͑ ͒ cooling in the idle state only HX on we can prepare an ancilla qubit 2 in the state (͉0͘ϩ͉1͘)/ͱ2. Then the total state is a͉00͘ϩb͉10͘ϩa͉01͘ϩb͉11͘. Letting the Josephson- Ϫ ␾ ␾ gate e i Z1Z2/2 act on this state, we obtain a͉00͘ϩei b͉10͘ ␾ Ϫ ␾ ␾ ϩei a͉01͘ϩb͉11͘ϰe i Z1/2͉␺͉͘0͘ϩei Z1/2͉␺͉͘1͘. We then ϩ ͑ measure Z2.Ifwefind 1 with probablity 1/2) then the Ϫ ␾ i Z1/2͉␺͉͘ ͘ state has collapsed to e 0 , which is the required FIG. 1. Gate teleportation of single-qubit operation Rz . Initially operation on qubit 1. If we find Ϫ1 then the state is Alice has ͉␺͘ and ͉0͘. Bob has ͉1͘. Time proceeds from left to i␾Z /2 1 e 1 ͉␺͉͘1͘, which is an erred state. To correct it we apply right. Starting from the three-qubit state ͉␺͉͘01͘, the task is to Ϫ ␾ i Z1Z2 ͉␺ the pulse e , which takes the erred state to the correct obtain Rz ͘. The protocol shown succeeds with probability 1/2. state When it fails the operation R† is applied instead. Fractions give the Ϫi␾Z /2 z Ϫe 1 ͉␺͉͘1͘. We then reinitialize the ancilla qubit. This probability of a branch; 0 and 1 in a gray box are possible measure- method for implementing Zi succeeds with certainty after ment outcomes of the observable in the preceding gray box. See one measurement, possibly requiring ͑with probability 1/2) text for full details. one correction step. ͑we use units where បϭ1), where ␸␣ϭ͐tdtЈJ␣(tЈ), and we III. EXCHANGE BASED PROPOSALS have suppressed the qubit indices for clarity. In preparation We now turn to the QC proposals based on exchange of our main result, we first prove the following: Gϭ ␸Ќ ␸z ϵ ␲ interactions, e.g., Refs. ͓2–4,12͔. In these systems, that are Proposition. The set ͕Uij( , ), R j␤ exp(i / ␴␤ ͖ ␤ϭ some of the more promising candidates for scalable QC, the 4 j ) ( x,z) is universal for quantum computation. qubit-qubit interaction can be written as an axially symmet- Proof. A set of continuous one-qubit unitary gates and any ric exchange interaction of the form two-body Hamiltonian entangling qubits are universal for ͓ ͔ ex͑ ͒ϭ Ќ͑ ͒͑ ϩ ͒ϩ z ͑ ͒ ͑ ͒ quantum computation 15 . The exchange Hamiltonian Hij t Jij t XiX j Y iY j Jij t ZiZ j , 1 ex Hij clearly can generate entanglement, so it suffices to ␣ ␣ϭЌ where Jij(t)( ,z) are controllable coupling constants. show that we can generate all single-qubit transfor- z ϭ Þ G The XY (XXZ) model is the case when Jij 0( 0). The mations using . Two of the Pauli matrices are z ϭ Ќ ␴␤ϭϪ 2 ␪ ؠ ␸ Heisenberg interaction is the case when Jij(t) Jij(t). See given simply by j iRj␤ . Now, let CA exp(i B) Ref. ͓5͔ for a classification of various QC models by the type ϵexp(Ϫi␪A)exp(i␸B)exp(ϩi␪A); two useful identities for of exchange interaction. In agreement with the QC proposals anticommuting A,B with A2ϭI ͑the identity͒ are ͓ ͔ Ќ 2–4,12 , we assume here that Jij(t) is competely control- lable and allow that Jz (t) may not be controllable.
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