Quantum Control Classical Input Classical Output QUANTUM WORLD QUANTUM INFORMATION INSIDE Preparation Readout Dynamics

Q.C. Paradigms Hilbert Space Yes

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). Unary system D = 4 Subsystem division 2 qubits ; D = 4

Example: Rydberg atom

We don’t live in Hilbert space 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. What physical resources are required to achieve a Hilbert-space dimension sufficient to carry out a given calculation? quant-ph/

Hilbert space and physical resources Hilbert-space dimension is a physical quantity that costs physical resources. Single degree of freedom Action quantifies the physical resources. Planck’s constant sets the scale.

Hilbert space and physical resources Primary resource is Hilbert-space dimension. Hilbert-space dimension costs physical resources. Many degrees of freedom Hilbert-space dimension measured in qubit units. Identical degrees of freedom Number of degrees of freedom qudits Strictly scalable resource requirement Scalable resource requirement

Hilbert space and physical resources Primary resource is Hilbert-space dimension. Hilbert-space dimension costs physical resources. Many degrees of freedom x 3, p 3 x 2, p 2 x 1, p 1 x, p

Quantum computing in a single atom Characteristic scales are set by “atomic units” LengthActionEnergyMomentum Bohr Hilbert-space dimension up to n 3 degrees of freedom

Quantum computing in a single atom Characteristic scales are set by “atomic units” LengthActionEnergyMomentum Bohr 5 times the diameter of the Sun Poor scaling in this physically unary quantum computer

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? 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. DiVincenzo’s criteria DiVincenzo, Fortschr. Phys. 48, 771 (2000)

Physical resources: classical vs. quantum A few electrons on a capacitor A pit on a compact disk A 0 or 1 on the printed page A smoke signal rising from a distant mesa Classical bit Quantum bit A classical bit involves many degrees of freedom. Our scaling analysis applies, but with a basic phase-space scale of arbitrarily small. Limit set by noise, not fundamental physics. The scale of irreducible resource requirements is always set by Planck’s constant. An electron spin in a semiconductor A flux quantum in a superconductor A photon of coupled ions Energy levels in an atom

Why Atomic Qubits? State Preparation Initialization Entropy Dump State Manipulation Potentials/Traps Control Fields Particle Interactions State Readout Quantum Jumps State Tomography Process Tomography Fluorescence Laser coolingQuantum Optics NMR

Optical Lattices

Designing Optical Lattices  ij  1 3  0 2   i  ijk  k  /2 1/2  1/2  3/2 1/2  1/2 S 1/2 P 3/2 Tensor Polarizability Effective scalar + Zeeman interaction

Lin-  -Lin Lattice

Multiparticle Control Controlled Collisions

Dipole-Dipole Interactions Resonant dipole-dipole interaction  tot  dd  2  (Quasistatic potential) (Dicke Superradiant State) V dd ~ d 2 r 3    ~ d 2  3 Figure of Merit

Cooperative level shift e 1 e 2 g 1 g 2 e 1 g 2 g 1 e 2 Bare     e 1 e 2 g 1 g 2 V dd Coupled    g 1 g 2 e 1 e 2 Dressed

r 12 Two Gaussian-Localized Atoms

Three-Level Atoms Atomic Spectrum“Molecular” Spectrum

Molecular Hyperfine Atomic Spectrum 5P 1/2 5S 1/2 F=2 F=1 F=2 F=1 87 Rb 0.8 GHz 6.8 GHz “Molecular” Spectrum Brennen et al. PRA (2002)

Resolvability = Fidelity   E 11  E 00  2E 01   ij   E c   Figure of Merit: Controlled-Phase Gate Fidelity

Leakage: Spin-Dipolar Interaction azimuthally symmetric trap 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”.

Energy Spectrum

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?

Carl Caves (UNM), Robin Blume-Kohout (LANL) Gavin Brennen (UNM/NIST), Poul Jessen (UA), Carl Williams (NIST) I.H. Deutsch, Dept. Of Physics and Astronomy University of New Mexico Collaborators: Physical Resource Requirements for Scalable Q.C. Quantum Logic via Dipole-Dipole Interactions René Stock (UNM), Eric Bolda (NIST) Quantum Logic via Ground-State Collisions