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

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

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

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

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

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

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

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

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

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

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

1. Fault tolerant quantum computing

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

1. Fault tolerant quantum computing Stabilizer codes

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

Fault tolerant quantum computing - Fault Tolerance Assessment-

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

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

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

2. Motivation: a critical view Structural redundancy

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

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

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

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

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

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

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

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

4. The quantum configuration Correction circuit with rQGA

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

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)

4. The quantum configuration Correction circuit with rQGA Configuration state

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

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

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

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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