1. 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

2. Presentation Outline
Fault tolerant quantum computing: a brief presentation
Motivation: a critical view
The rQHW (rQGA) solution
The quantum configuration
Qualitative assessment (accuracy threshold)
Conclusions

3. 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

4. 1. Fault tolerant quantum computing
ERROR TYPES
Bit-flip
Phase shift
Small amplitude errors
Similar to analog errors

5. 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

6. 1. Fault tolerant quantum computing
Phase-shift error backward propagation

7. 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

8. 1. Fault tolerant quantum computing
Strategies for attaining Fault Tolerance
Ancilla and syndrome accuracy for FT recovery
Error detection and correction by appropriate encoding

9. 1. Fault tolerant quantum computing
Error Detection and Correction Codes (Steane)

10. 1. Fault tolerant quantum computing
Error Detection and Correction Codes (Steane)
Steane Encoding Circuit

11. 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

12. 1. Fault tolerant quantum computing

13. 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

14. 1. Fault tolerant quantum computing
Stabilizer codes

15. 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 steps
No-coding case
For real cases (Shor’s algorithm) the accuracy threshold is ~ 10-4

16. Fault tolerant quantum computing - Fault Tolerance Assessment-

17. 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

18. 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

19. 2. Motivation: a critical view
Ancilla correction
(ad infinitum ?)

20. 2. Motivation: a critical view
Structural redundancy

21. 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

22. 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

23. 3. The rQHW (rQGA) solution
The analysis provided in the critique section suggests the cure: rQHW
The rQHW concept was already addressed [Nielsen & Chuang]

24. 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

25. 3. The rQHW (rQGA) solution
rQGA structure: limitations consequence

26. 3. The rQHW (rQGA) solution
Appropriate gates [Barenco et. al]

27. 3. The rQHW (rQGA) solution (basic cell)

28. 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

29. 4. The quantum configuration
Correction circuit with rQGA

30. 4. The quantum configuration
rQGA for correction circuit (Stabilizer coding + Steane ancilla)

31. 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 12-qubit 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)

32. 4. The quantum configuration
Correction circuit with rQGA
Configuration state

33. 5. Qualitative assessment
Accuracy Threshold Analysis
Performed as prescribed by John Preskill
Assumes correct preservation of the configuration register
Overall error rate
Accuracy threshold

34. 5. Qualitative assessment
S =2; fr =1/4; we consider a high p =6.
Technological accuracy limit is provided for comparison

35. 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

36. 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)

37. Thank You