Transcript Document

Improving Quantum Circuit Dependability
with Reconfigurable Quantum Gate Arrays
Mihai Udrescu Lucian Prodan Mircea Vlăduţiu
Advanced Computing Systems and Architectures Laboratory
Computer Engineering Department
University “Politehnica” Timişoara, Romania
University “Politehnica” Timişoara
Presentation Outline
1. Fault tolerant quantum computing: a brief
presentation
2. Motivation: a critical view
3. The rQHW (rQGA) solution
4. The quantum configuration
5. Qualitative assessment (accuracy threshold)
6. Conclusions
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
• Dependability is vital in QC
• The errors are ubiquitous
• The main enemy: decoherence
– i.e. the quantum state (a microscopically encoded
superposition of classical states) is measured by the
macroscopic environment
• The error model [Preskill] is probabilistic and
assumes errors that are:
– Single
– Non-correlated
– Store errors, gate errors
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1. Fault tolerant quantum computing
• ERROR TYPES
– Bit-flip 0
1,
1
0
error
  a0 0  a1 1 
 a0 1  a1 0
– Phase shift
0
0 ,
1
1
error
  a0 0  a1 1 
 a0 0  a1 1
– Small amplitude errors
•Similar to analog errors
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1. Fault tolerant quantum computing
• QC constraints
– The observation destroys the state
– Information copy is impossible
• QC additional problems
– We need to be able to get state information without
destroying it => we are forced to use ancilla qubits
– We need a fault tolerant recovery process, otherwise
the coding fault tolerant techniques become useless
– The phase-shift error propagates backward
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1. Fault tolerant quantum computing
• Phase-shift error backward propagation
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1. Fault tolerant quantum computing
• Strategies for attaining Fault Tolerance
– Digitizing small errors
– Using ancilla qubits in order to measure the information
without destroying it
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
• Strategies for attaining Fault Tolerance
– Ancilla and syndrome accuracy for FT recovery
– Error detection and correction by appropriate encoding
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
• Error Detection and Correction Codes (Steane)
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
• Error Detection and Correction Codes (Steane)
Steane Encoding Circuit
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1. Fault tolerant quantum computing
• Error Detection and Correction Circuit (Steane)
– Works with Steane codes
– Ancilla encoding according to Steane’s procedure
– Implementation according to the strategies for
attaining fault tolerance
– In order to obtain fault tolerant (safe) recovery,
structural redundancy is employed
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
University “Politehnica” Timişoara
1. Fault tolerant quantum computing
• Stabilizer codes
– Generalization of Steane 7-qubit encoding
– Has a special formalism [D. Gottesman]
– Any new stabilizer code can be obtained by permuting
Hamming matrix columns
– Special gates for manipulating these codes were developed
Stabilizer generator
collection
Stabilizer code
check matrix
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1. Fault tolerant
quantum
computing
• Stabilizer codes
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1.
Fault tolerant quantum computing
- Fault Tolerance Assessment-
• Accuracy threshold: the fault rate that still allows the
overall correct computation
• [Preskill]: for a quantum code that corrects r errors
with a methodology that requires rp computational
p
steps
  log N 
• No-coding case
 N 1
• For real cases (Shor’s algorithm) the accuracy
threshold is ~ 10-4
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1.
Fault tolerant quantum computing
- Fault Tolerance Assessment-
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1. Fault tolerant quantum computing
- Fault Tolerance Assessment-
• Arbitrary long Fault Tolerant Quantum Computation
• Threat = not enough correction steps => r+1
errors accumulating before correction
• The solution: concatenated coding
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2. Motivation: a critical view
• The big picture
– In QC the circuits are prone to frequent failures
– Safe recovery is a problem
– A successful FTAM (for our error model –
single random fault) means that, for a x fault
rate, the overall circuit error rate is x 2
– Besides coding, structural redundancy is
employed
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2. Motivation: a critical view
Ancilla correction
(ad infinitum ?)
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2. Motivation: a critical view
Structural redundancy
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2. Motivation: a critical view
• Issues to be settled
– The fault occurrence model has not taken into
account the correlated errors
– The inflexibility of ancilla qubit preparation,
requires that al least 2 sets of ancilla is prepared
even if the first one is correct
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2. Motivation: a critical view
• Correlated errors – destructive for
concatenated coding
Steane’s 7 qubit code on 3
concatenated levels:
5 faults from 343 qubits
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3. The rQHW (rQGA) solution
• The analysis provided in the critique section
suggests the cure: rQHW
• The rQHW concept was already addressed
[Nielsen & Chuang]
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3. The rQHW (rQGA) solution
• Limitations for reconfigurable (programmable) Quantum
Gate Arrays
– The gate array must operate “in a probabilistic fashion”
[Nielsen & Chuang] in order to perform any unitary operation
– It is impossible to build a switch-based rQGA
– Consequence of cloning impossibility
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3. The rQHW (rQGA) solution
• rQGA structure: limitations consequence
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3. The rQHW (rQGA) solution
Appropriate gates [Barenco et. al]
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3. The rQHW (rQGA) solution (basic cell)
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4. The quantum configuration
• Code Generation with rQGA
– A classical configuration register for each distinct
stabilizer code
– When the configurations for all possible 7-qubit
stabilizer generated codes are superposed in a
quantum state, then the rQGA is the superposition
of all 7-qubit stabilizer encoder circuits
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4. The quantum configuration
• Correction circuit
with rQGA
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4. The quantum configuration
• rQGA for correction circuit (Stabilizer coding + Steane ancilla)
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4. The quantum configuration
Reconfigurable Quantum Hardware
• 2 Basic cells used
• The configuration register can be reduced to a
classical register which is non-entangled with a 12qubit quantum state
• The configuration state corresponds to a superposition
of allowed stabilizer codes (obtained by permuting the
columns of HA Hamming matrix)
• Not all allowed stabilizer circuits are generated
because it is not a power of 2 number (configuration
state is hard to generate)
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4. The quantum configuration
• Correction circuit with rQGA
– Configuration state
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5. Qualitative assessment
• Accuracy Threshold Analysis
– Performed as prescribed by John Preskill
– Assumes correct preservation of the configuration
register
– Overall error rate

4
2
3

8
3
– Accuracy threshold

rQHW
threshold
 log N 
1
 p 1 f r 
S
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5. Qualitative assessment
• S =2; fr =1/4; we consider a high p =6.
• Technological accuracy limit is provided for
comparison
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6. Conclusions
• Valid FTAM techniques means an overall x 2 failure rate for a qubit
and gate failure rate of the order x
• The rQGA technique reduces the gate error problem to preserving
a correct configuration state
• This state is simplified for the example correction circuit (stabilizer
coding + Steane ancilla)
• The quantum configuration is used in order to dictate a
superposition of distinct correcting circuits. The configuration
register is measured => just one of the circuits (corresponding to
the measured configuration) is used for the actual correction.
• k superposed correcting circuits, x error rate/gate => x k overall
error rate
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6. Conclusions
• The accuracy threshold is improved, as it clear
dominates the graphical representation of the
technological limit
• The rQGA strategy may replace concatenated coding (a
technique that may be useless in the presence of
correlated errors)
• Future work is aiming at
– Defining the framework for developing evolvable quantum
circuits (EHW = RHW + GA)
– Quantitative assessment of Accuracy Threshold by simulation
(Simulated Fault Injection)
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Thank You