QUANTUM WORLD Quantum Control Classical Input Preparation Dynamics Readout Classical Output QUANTUM INFORMATION INSIDE
Hilbert spaces are fungible ADJECTIVE: 1. Law.Returnable or negotiable in kind or by substitution, as a quantity of grain for an equal amount of the same kind of grain. 2. Interchangeable. ETYMOLOGY: Medieval Latin fungibilis, from Latin fung (vice), to perform (in place of). Subsystem division 2 qubits; D = 4 Unary system D = 4
Example: Rydberg atom http://gomez.physics.lsa.umich.edu/~phil/qcomp.html
A Hilbert space is endowed with structure by the physical system described by it, not vice versa. The structure comes from preferred observables associated with spacetime symmetries that anchor Hilbert space to the external world. Hilbert-space dimension is determined by physics. The dimension available for a quantum computation is a physical quantity that costs physical resources. Key Question What physical resources are required to achieve a Hilbert-space dimension sufficient to carry out a given calculation? quant-ph/0204157 We don’t live in Hilbert space
Action quantifies the physical resources. Planck’s constant sets the scale. Hilbert space and physical resources Hilbert-space dimension is a physical quantity that costs physical resources. Single degree of freedom
Hilbert space and physical resources Primary resource is Hilbert-space dimension. Hilbert-space dimension costs physical resources. Many degrees of freedom Number of degrees of freedom Identical degrees of freedom Hilbert-space dimension measured in qubit units. Scalable resource requirement Strictly scalable resource requirement qudits
x3, p3 Primary resource is Hilbert-space dimension. Hilbert-space dimension costs physical resources. x2, p2 x1, p1 x, p Hilbert space and physical resources Many degrees of freedom
Length Momentum Action Energy Bohr 3 degrees of freedom Hilbert-space dimension up to n Quantum computing in a single atom Characteristic scales are set by “atomic units”
Length Momentum Action Energy Bohr Poor scaling in this physically unary quantum computer Quantum computing in a single atom Characteristic scales are set by “atomic units” 5 times the diameter of the Sun
Other requirements for a scalable quantum computer Avoiding an exponential demand for physical resources requires a quantum computer to have a scalable tensor-product structure. This is a necessary, but not sufficient requirement for a scalable quantum computer. Are there other requirements? DiVincenzo’s criteria DiVincenzo, Fortschr. Phys. 48, 771 (2000) 1. Scalability:A scalable physical system with well characterized parts, usually qubits. 2. Initialization:The ability to initialize the system in a simple fiducial state. 3. Control: The ability to control the state of the computer using sequences of elementary universal gates. 4. Stability:Long decoherence times, together with the ability to suppress decoherence through error correction and fault-tolerant computation. 5. Measurement:The ability to read out the state of the computer in a convenient product basis.
Classical bit A fewelectronson a capacitor Anelectron spinin a semiconductor Apiton a compact disk Afluxquantumin a superconductor A0or1on the printed page Aphotonof coupled ions A smoke signalrising from a distant mesa Energy levels in an atom Quantum bit Physical resources: classical vs. quantum A classical bit involves many degrees of freedom. Our scaling analysis applies, but with a basic phase-space scale ofarbitrarily small. Limit set by noise,not fundamental physics. The scale of irreducible resource requirements is always set by Planck’s constant.
State Preparation • Initialization • Entropy Dump Why Atomic Qubits? State Manipulation • Potentials/Traps • Control Fields • Particle Interactions Laser cooling Quantum Optics NMR State Readout • Quantum Jumps • State Tomography • Process Tomography Fluorescence
Designing Optical Lattices Tensor Polarizability P3/2 3 / 2 - 1 / 2 1 / 2 - 3 / 2 1 ( ) 1 2 2 2 i a = - a d + e s 1 1 ij 0 ij ijk k 3 3 3 3 S1/2 1 / 2 1 / 2 - Effective scalar + Zeeman interaction
Multiparticle Control Controlled Collisions
+ + - - 2 G = G ¢ + G £ G ¢ tot dd 2 d 2 d V ~ h ~ G ¢ dd 3 r 3 D (Quasistatic potential) (Dicke Superradiant State) Figure of Merit Dipole-Dipole Interactions • Resonant dipole-dipole interaction
Coupled Bare Dressed ¢ e e e e e e 1 2 1 2 1 2 V y dd e g g e - ¢ 1 2 1 2 y + D y + ¢ g g 1 2 g g g g 1 2 1 2 Cooperative level shift
r 12 Two Gaussian-Localized Atoms
Atomic Spectrum “Molecular” Spectrum Three-Level Atoms
“Molecular” Spectrum Brennen et al. PRA 65 022313 (2002) Molecular Hyperfine Atomic Spectrum 0.8 GHz F=2 5P1/2 F=1 F=2 6.8 GHz 5S1/2 F=1 87Rb
Figure of Merit: E E 2 E E + - D 11 00 01 c k = = h h G G ij ij Resolvability = Fidelity Controlled-Phase Gate Fidelity
azimuthally symmetric trap Leakage: Spin-Dipolar Interaction Noncentral force
Suppressing Leakage Through Trap Energy and momentum conservation suppress spin flip for localized and separated atoms.
Dimer Control • Lattice probes dimer dynamics • Localization fixes internuclear coordinate
Separated-Atom Cold-Collision Short range interaction potential, well characterized by a hard-sphere scattering with an “effective scattering length”.
Shape Resonance Molecular bound state, near dissociation, plays the role of an auxiliary level for controlled phase-shift.
Dreams for the Future • Qudit logic: Improved fault-tolerant thresholds? • Topological lattice - Planar codes?
http://info.phys.unm.edu/~deutschgroup I.H. Deutsch, Dept. Of Physics and Astronomy University of New Mexico • Collaborators: • Physical Resource Requirements for Scalable Q.C. Carl Caves (UNM), Robin Blume-Kohout (LANL) • Quantum Logic via Dipole-Dipole Interactions Gavin Brennen (UNM/NIST), Poul Jessen (UA), Carl Williams (NIST) • Quantum Logic via Ground-State Collisions René Stock (UNM), Eric Bolda (NIST)