PCE STAMP Physics & Astronomy UBC Vancouver Pacific Institute for Theoretical Physics

MISSION of PITP http://pitp.physics.ubc.ca/index.html PITP is an international institute, funded internationally, with an international mission- to bring together groups of high-quality researchers, from around the world, and foster path-breaking new research in all branches of theoretical physics. Recognizing that theoretical physics is central to the whole of science, PITP fosters links to other subjects, including chemistry & biology, and tries to ‘seed’ new developments in these areas. Recognizing the decisive influence that physics and other sciences have in the modern world, PITP will advise and assist in areas within its competence, and provide information to the general public about theoretical physics and related topics. http://pitp.physics.ubc.ca/index.html

COHERENCE WINDOWS in IINSULATING MAGNETS Pacific Institute for Theoretical Physics Work done with I.S. Tupitsyn (PITP) G. Milburn, A. Hines (Queensland) T. Stace (Cambridge, UK) http://pitp.physics.ubc.ca/index.html

H = Sj ( Dj tjx + ej tjz ) + Sij Vij tiz tjz QUBIT NETWORKS We are interested in networks of quantum gates made from solid-state qubits, typically described as 2-level systems: the general Hamiltonian of interest is H = Sj ( Dj tjx + ej tjz ) + Sij Vij tiz tjz Dj(t), ej(t), Vij (t) At least some of the ‘control parameters’ can be varied, to make gate operations. In most solid-state qubit designs this is done by carrying out sequences of pulses on individual qubits. One can also look at more sophisticated designs (adiabatic quantum computing, topological quantum computing), but no practical implementation has yet emerged.

SOLID-STATE QUBITS: Theoretical Designs & Experiments Here are a few: (1) Superconducting SQUID qubits (where qubit states are flux states); all parameters can be controlled. (2) Magnetic molecule qubits (where an easy axis anisotropy gives 2 low energy spin states, which communicate via tunneling, and couple via exchange or dipolar interactions. Control of individual qubit fields is easy in principle- interspin couplings less so... (3) Spins in semiconductors (or in Q Dots). Local fields can be partially controlled, & the exchange coupling is also controllable.

PROBLEM #1: The MECHANISMS of Decoherence For many people the most fundamental problem facing any kind of quantum information processing (QUIP) system, in which entanglement between a large number of quantum variables is used as a resource, is that of Decoherence. The problem is that the QUIP variables will usually become rapidly entangled with environmental variables (ie., all other degrees of freedom which we cannot track, but which couple to the QUIP system). Averaging over these unknown variables effectively ‘smears’ the delicate phase relations encoding the quantum information. Far from being an irritating problem of ‘dirt’, the problem of decoherence raises interesting fundamental questions- in particular: Is decoherence an ineluctable feature of QUIP? And how does it increase with the level of entanglement (eg., with the number N of entangled qubits) ? (2) Does decoherence go away if we take the temperature T -> 0 ? To answer any of these questions we actually need to answer another question first: What are the mechanisms which cause decoherence in the 1st place? This turns out to be a very interesting physics problem.

. ~. ~ ~. ~” ~.: ~`”: ~`/: . . ,’` LOCALISED `’~.,`.,’ ..’` ..: .’` WHAT ARE THE LOW- ENERGY EXCITATIONS IN A SOLID ? DELOCALISED Phonons, photons, magnons, electrons, ……… LOCALISED Defects, Dislocations, Paramagnetic impurities, Nuclear Spins, ……. . . ………………….. . `’~.,`.,’ ..’` . ~. ~ ~. ~” ~.: At right- artist’s view of energy distribution at low T in a solid- at low T most energy is in localised states. INSET: heat relaxation in bulk Cu at low T ~`”: ~`/: . . ..: .’` ,’` .`, .’`* .’,

H = Sj ( Dj tjx + ej tjz ) + Sij Vij tiz tjz PROBLEM #2: The DYNAMICS of Decoherence The theoretical problem is to calculate the dynamics of the “M-qubit” reduced density matrix for the following Hamiltonian, describing a set of N interacting qubits (with N > M typically): H = Sj ( Dj tjx + ej tjz ) + Sij Vij tiz tjz + Hspin({sk}) + Hosc({xq}) + int. The problem is to integrate out the 2 different environments coupling to the qubit system- this gives the N-qubit reduced density matrix. We may then average over other qubits if necessary to get the M-qubit density matrix operator rNM({tj}; t) The N-qubit density matrix contains all information about the dynamics of this QUIP (QUantum Information Processing) system- & all the quantum information is encoded in it. A question of some theoretical interest is- how do decoherence rates in this quantity vary with N and M ?

DECOHERENCE DYNAMICS from an EFFECTIVE H Consider the following Heff : H (Wo) = { [Dt+ exp(-i Sk ak.sk) + H.c.] + eotz (qubit) + tz wk.sk + hk.sk (bath spins) + inter-spin interactions At first glance a solution of this seems very forbidding. However it turns out one can solve for the reduced density matrix of the central spin in all interesting parameter regimes- & the decoherence mechanisms are easy to identify: (i) Noise decoherence: Random phases added to different Feynman paths by the noise field. (ii) Precessional decoherence: the phase accumulated by bath spins between qubit flips. (iii) Topological Decoherence: The phase induced in the bath spin dynamics by the qubit flip itself USUALLY PRECESSIONAL DECOHERENCE DOMINATES Precessional decoherence Noise decoherence source This leads to the very interesting result that one can have decoherence dominated by processes which cause little or no dissipation

Candidates for Magnetic Qubits One of the candidates discussed for quantum computations is magnetic systems. Note that very large magnetic domain walls have already shown macroscopic tunneling, just like SQUID flux. Right now interest is focussed on magnetic molecules and ions which behave as 2-level systems- as ‘Qubits”. Ho ions in LiYF4 host

The Fe-8 MOLECULE Low-T Quantum regime- effective Hamiltonian (T < 0.36 K): Longitudinal bias: Fe8 S = 10 Eigenstates: Which also defines orthonormal states: Feynman Paths on the spin sphere for a biaxial potential. Application of a field pulls the paths towards the field

HYPERFINE COUPLING to spin bath (NUCLEAR SPINS) Define the set of fields: Static component is: Some of the couplings in Fe-8 (at H=0) Component which flips is: This gives a ‘central spin’ Hamiltonian:

Structure of NUCLEAR MULTIPLET There are 215 nuclear sites in the molecule Transitions between states of different total polarisation (T1 process) driven mainly by molecular tunneling) Total width of gaussian multiplet: (NB: This decreases with increasing applied field) For Fe-8 at H=0, width is ~7 mK (depends on isotopic concentrations) Coupling to PHONONS Effective coupling to qubit:

DECOHERENCE Fe-8 Molecule At low applied transverse fields, decoherence switches on very fast- we expect incoherent spin relaxation: DECOHERENCE in the Fe-8 Molecule However, at high fields, system can be in coherence window, in which qubit dynamics is too fast for nuclear spins to follow, but still much slower than phonons This frequency window we call the COHERENCE WINDOW- typically

DECOHERENCE in the LiHo system Here the dominant hyperfine coupling is to a single Ho nuclear spin. So to get high coherence we need OPTIMUM RESULTS: For more details: Stamp, P.C.E., Tupitsyn, I.S., “Coherence window in the dynamics of quantum nanomagnets”, Phys Rev B69, 014401 (2004) Stamp, P.C.E., “Phase Dynamics of Solid-State Qubits: Magnets and Superconductors”, J. Quantum Computers & Computing 4, 20-62 (2003)

DECOHERENCE in Superconducting Qubits The oscillator bath (electrons, photons, phonons) decoherence rate: tf-1 ~ Do g(D,T) coth (D/2kT) (Caldeira-Leggett). This is often many orders of magnitude smaller than the experimental decoherence rates. The spin bath decoherence will be caused by a combination of charge & spin (nuclear & paramagnetic) defects- in junction, SQUID, and substrate. 1/tf = Do (Eo/8D0)2 The basic problem with any theory-experiment comparison for these systems is that most of these 2-level systems are basically just junk (coming from impurities and defects), whose characteristics are hard to quantify. However this may change with recent work from the groups of Nakamura, Van Harlingen, and Martinis. Currently 8 groups have seen coherent oscillations in superconducting qubits, and 2 have seen entanglement between qubit pairs.

WARNING: 3rd PARTY DECOHERENCE This is fairly simple- it is decoherence in the dynamics of a system A (coordinate Q) caused by indirect entanglement with an environment E- the entanglement is achieved via a 3rd party B (coordinate X). Consider the 2-slit expt with buckyballs. The COM coordinate Q of the buckyball does not couple directly to the vibrational modes {qk } of the buckyball- by definition. However BOTH couple to the slits in the system, in a distinguishable way. Note: the state of the 2 slits, described by a coordinate X, is irrelevant- it does not need to change at all. We can think of it as a scattering potential, caused by a system with infinite mass (although recall Bohr’s response to Einstein, which includes the recoil of the 2 slit system). It is a PASSIVE 3rd party. Ex: Buckyball decoherence ACTIVE 3rd PARTY: Here the system state correlates with the 3rd party, which then goes on to change the environment to correlate with Q. We can also think of the 3rd party X as PREPARING the states of both system and environment. Alternatively we can think of the system and the environment as independently measuring the state of X. In either case we see that system and environment end up being correlated/entangled. Note the final state of X is not necessarily relevant- it can be changed in an arbitrary way after the 2nd interaction of X. Thus X need not be part of the environment. Note we could also have more than one intermediary- ie., X, Y, etc.- with correlations/entanglement are transmitted along a chain (& they can wiped out before the process is finished).

CONCLUSIONS (1) The Decoherence Mechanisms in solid-state qubits can be understood using 2 different ‘universality classes’ of model. At low T the principal mechanism of decoherence is that coming from discrete states (eg., 2-level systems) in the environment. The decoherence from these systems does NOT in general go to zero as T -> 0. In solid-state qubits there typically exists a ‘coherence window’ in energy space- if the system of qubits is operated in this energy window the decoherence rate can be very small. Magnetic solid-state qubits look like a good bet- their intrinsic decoherence rate can be made much lower than that of superconductors. The problem so far, which looks like being solved pretty soon, is to read and write on these qubits INDIVIDUALLY. The theoretical problem of how decoherence increases with the number of entangled qubits remains interesting. A similar problem can be formulated for other kinds of Q Computing algorithm (eg., adiabatic Q Computation)