Universal Quantum Interfaces1

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Universal Quantum Interfaces1 and Quantum
Recursive Algorithms2
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Brian Jacokes, TJHSST Julia Kempe, LRI In
progresscollaboration welcome
Seth Lloyd, MIT Jean-Jacques Slotine,
MIT quant-ph/0303048
HP/MIT WorkshopCorvallis, OR 8/13/03
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Entering the Era of Quantum Control
Cavity QED
Liquid NMR
Quantum Dot
Ion Trap
Lattice BEC
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Promises of Full Quantum Control
011010110101101110100101 -------------------- 110
100010000
Computation
Teleportation
Cryptography
Simulation
Communication
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Missing Roadmap!
How much control is enough?

• Full control of one qubit.
• Generic coupling to a closed quantum system.

Quantum Interface
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Interfaces
Environment
System
Observer
Controller
Computer
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Interfaces
System
Interface
Computer
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Outline of construction
Given
Fixed generic interaction
Closed-system Hamiltonian
Projective spin measurement
Arbitrary qubit Hamiltonian
Outline

• Fixed interaction Arbitrary interaction
• Coherent system control
• Any Hermitian two-outcome measurement
• Any two-outcome measurement
• Any generalized measurement
• Any quantum operation

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Universality
Kraus Representation Theorem
Any linear, completely positive map
between density matrices may be written as
QuantumOperation
where
Church of the Larger Hilbert Space
unnormalized
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Fixed interaction Any Interaction
Key formulas
Generically generate full Lie
Algebra on
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Coherent system control
Simple, given arbitrary interactions
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Any Hermitian two-outcome measurement

• Prepare interface in state
• Evolve via Hamiltonian
• Measure interface.

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Any two-outcome measurement
Apply feedback
Polar decomposition theorem
Any operator can be written asfor some unitary
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Any generalized measurement
Build up iteratively
If is obtained, measure
Etc.
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Any quantum operation
Discard (forget) measurement outcomes
The single qubit interface is universal.
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Efficiency Quantum Computation
Let d dim
O(d 2) basis of unitaries O(d 2) time to
simulate any one
Efficient quantum algorithms d2n, O(polylog d )
sized circuits
Need O(n ) Universal Quantum Interfaces (UQIs)!
A) Quantum Logic

• Connect qubits pairwise by UQIs
• Measure UQIs to decouple qubits
• O(n ) SWAPs to bring qubits adjacent
• O(1) time to effect a 2-qubit gate
• 2-qubit gates are universal.

B) State Preparation Measurement
Decouple qubits and selectively measure, rotate
qubits
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Candidate UQIs

• 1 Electromagnetic mode in optical cavity
• 1 Ion in an ion trap
• 1 Nuclear spin in a molecule
• 1 Flux/charge qubit in a superconducting circuit
• 1 Optically active site on a molecule

A degree of freedom can be tailored to be a UQI!
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Open Questions
Open quantum systems

• Use of UQIs for decoupling open quantumsystems
  from their environment? (DFS/QEC)

Efficiency

• Quantify controllability/observability of
  openquantum systems by UQIs.

Networkability

• O(1) UQIs for quantum computation withadditional
  controls?

Read more atquant-ph/0303048
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• Ideas for future talks
• Discuss spin chain quantum computation
• Show control problem as an integral
• Go over commutator example, showing how to get a
  complete set of generators
• Animation of wiggling one end to compute
• Add colors to formulas, figures
• Discuss robustness, error propagation
• Compare simulated time, real time

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Quantum Recursion Ordered Search
Query Complexity
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Invariant Algorithms
query algorithm
Computer search
( )
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Universal Curve?
3 queries, 53 items
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Game Tree Evaluation
Query Complexity
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Game Tree Evaluation
Classical Randomized Algorithm

• Evaluate left/right subtree based on coin flip
• If 0, return 1. Otherwise evaluate other subtree.

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Base cases
Blue Classical
NAND(1)
Black Quantum
Queries Success Probability
Queries Success Probability
NAND(2)
Queries Success Probability
Queries Success Probability
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Noisy Quantum Recursion
HØyer et al., 2003 (quant-ph/0304052)
Perfect oracle
Noisy oracle
Naïve recursive Grover

• 1 query in size-4 list
• Recursion, majority voting ?

Adaptive Grover step
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Summary

• 1 qubit can be a universal I/O port (interface).
• Best-known quantum algorithm for searchingan
  ordered list is recursive.
• Quantum computers might be good at playing games.