� � � � � � � � � � � � � � � � � � � � � ���� ���� | M � • M X Photos placed in horizontal posi1on p fail with even amount of white space � ��� ���� ���� � � / • • � 0 M Z H Unencode between photos and header � � � � � � � � � � � ���� ���� � � � � / � M � 0 X Z p Fault-tolerant quantum computing Andrew J. Landahl 12/5/11 This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. QEC11: Second Interna1onal Conference on Quantum Error Correc1on
Fault-tolerance Defini:on: A process is fault‐tolerant if it achieves its task (arbitrarily well, efficiently) in the event of a failure. Example: (relevant for both classical and quantum informa:on) Task: Maintain data. (Sta1c: memory; moving: communica1on.) Failure: Data corrup1on. (Source: local environmental noise.) Process: Error correc1on. Solves by adding redundancy and processing. Fault tolerance: Can achieve failure probability ε by an O (log ( 1 / ε ))‐sized code as long as p < p c , the error threshold for the infinite code family. New Problem 1: The processing can itself fail! New Problem 2: How do we compute with data now that it is encoded? A fault‐tolerant compu&ng protocol maintains arbitrary computa1ons (arbitrarily well, efficiently) in the presence of faults to both its data and its processing. Central ques:ons: What combina1on of computa1on, control, and noise models admit fault‐tolerant computa1on? • What are the (minimal) resource costs for achieving fault‐tolerant quantum computa1on? • QEC11: Second Interna1onal Conference on Quantum Error Correc1on 2
Fault-tolerant classical computing Modern computers can calculate for weeks (years?) without failing: good hardware. • With faulty hardware: S1ll possible! (Cost, power, speed, etc . may drive us there.) • SeGng: Computa:onal model: Circuits in which each gate has exactly one output ( formulas ). • Control model: Parallel opera1on, refreshable bits allowed. • Noise model: Noisy gate = ideal gate followed by bit‐flip with probability p. • Approach: Simulate an ideal formula to desired precision ε using a faulty formula. • Encode data and process it with encoded gates which suppress error spreading. • Threshold Theorem: [von Neumann, 1952, 1956, 1966] A g ‐gate ideal formula can be simulated to precision ε by an O ( g log ( g / ε ))‐gate faulty formula as long as p < p c , • the accuracy threshold for classical computa1on. √ 2‐input gate accuracy threshold (1ght): • p c = (3 − 7) / 4 ≈ 8 . 9% [Unger, 2008] 2 k − 2 p c = 1 k →∞ (2 k +1)‐input gate accuracy threshold (1ght): • 50% 2 − − → k − 1 [Evans and Schulman, 2003; � � k k/ 2 − 1 / 2 Hajek and Weller, 1991] QEC11: Second Interna1onal Conference on Quantum Error Correc1on 3
Fault-tolerant quantum computing Computa:onal model • 1. Quantum Turing machine 2. Quantum walk H ( t = 0) 3. Adiaba1c H ( t = τ ) 4. Quantum circuit } Topological • The only model proven to admit fault‐tolerant quantum compu:ng Measurement‐based • Holonomic • Control model Noise model • • Parallel opera1on Nonincreasing error rate • • Necessary • Refreshable qubits Reliable classical computa1on • • Helpful • Fast classical computa1on No qubit leakage • • Convenient • Finite gate basis Uncorrelated noise • • Realis1c • Equal‐1me gates State‐independent noise • • 2D layout Uniformly faulty gates • • Local quantum processing • QEC11: Second Interna1onal Conference on Quantum Error Correc1on 4
Fault-tolerant quantum computing Task: Simulate an ideal quantum circuit with faulty components. (E.g., inverse quantum Fourier transform) | k 5 � • • • • • H | k 4 � π H • • • • 2 | k 3 � π π H • • • 4 2 | k 2 � π π π H • • 8 4 2 | k 1 � π π π π H • 16 8 4 2 | k 0 � π π π π π H 32 16 8 4 2 Four components of every known fault‐tolerant quantum compu:ng protocol (FTQCP) } 1. An infinite family of (stabilizer) quantum error‐correc1ng codes. FTQEC protocol 2. A protocol for extrac1ng the syndrome from each code. 3. A (classical) decoding algorithm for interpre1ng the syndrome of each code. } 4. A finite universal gate basis and a protocol for implemen1ng each gate in encoded form. Encoded computa:on Applica:on of an FTQCP to an ideal quantum circuit: Step 1: Compile the ideal ( g ‐gate) circuit into a new ( g’ ‐gate) circuit over the specified finite gate basis. Step 2: Choose a code large enough to suppress errors below O (1/ g’ ) with FTQEC. Step 3: Replace each qubit and gate (incl. prep. & meas.) in the new circuit with encoded (“logical”) versions. Step 4: Insert syndrome extrac1on aler each encoded gate. (Some may be able to be omimed.) Step 5: Insert decoding‐iden1fied recovery before each non‐Clifford encoded gate. Step 6: Aler final logical measurement, perform final classical decoding of outcomes to infer result. QEC11: Second Interna1onal Conference on Quantum Error Correc1on 5
Quantum compiling | k 5 � • • • • • H | k 4 � π • • • • H 2 | k 3 � π π • • • H 4 2 | k 2 � π π π • • H 8 4 2 | k 1 � π π π π • H 16 8 4 2 | k 0 � π π π π π H 32 16 8 4 2 Step 1: Compile the ideal circuit into a finite gate basis . The gate basis does not need to be physically • G realizable—we will simulate these “encoded” gates by physical gates, which may be different. Note: This “ quantum compiling ” is itself a simula1on, which may generate approximaEon errors • depending on the quantum circuit it simulates. If the ideal circuit has g gates, we want the approxima1on error to be ε = O ( 1 / g ). Dawson‐Nielsen Solovay‐Kitaev quantum compiling algorithm: • Finds ε ‐approxima1on to of length in 1me . O (log 3 . 97 (1 /ε )) • O (log 2 . 71 (1 / ε )) U gate Requires each gate to have its inverse in the finite gate basis. • g ′ = O ( g log 3 . 97 g ) Circuit to simulate now has gates. QEC11: Second Interna1onal Conference on Quantum Error Correc1on 6
Finite universal gate bases Popular choices: Nota1on legend: • S := Λ( i ) � 1 � 0 1 Λ( U ) := 2( | 0 � + e iθ | 1 � ) | θ � := √ Real gate basis: • { H, TOF } ∪ {| 0 � , M Z } 0 U T := Λ ( e iπ/ 4 ) | + � := | 2 π � | + i � := | π / 2 � CNOT := Λ( X ) TOF := Λ ( CNOT ) Kitaev gate basis: • { H, Λ( S ) } ∪ {| 0 � , M Z } | T � := | π/ 4 � √ H = ( X + Z ) / 2 Surface‐code cluster‐state gate basis: • { H, Λ ( Z ) } ∪ {| + � , M X , M Z } ∪ { M ( X ± Y ) / √ 2 } Standard gate basis: • { H, T, CNOT } ∪ {| 0 � , M Z } Overcomplete standard gate basis: • { I, X, Z, H, S, S † , T, T † , CNOT } ∪ {| 0 � , | + � , M X , M Z } Well‐suited to Solovay‐Kitaev quantum compiling. • G CSS � �� � Magic‐state standard gate basis: • { CNOT , | 0 � , | + � , M X , M Z } ∪{| + i �} ∪{| T �} � �� � G Clifford Very popular choice, but quantum compiling with this basis requires some finesse: • X , Z gates ignored: Classical update of “ logical Pauli frame .” • S , T , and H gates by the following circuits, which rely on “ magic states .” • Warning: not a Clifford circuit. ���� ���� | ψ � • S | ψ � | ψ � H | ψ � Z | ψ � • T | ψ � S X S S ���� ���� ���� ���� | + � • • S M Z | π/ 2 � M Z • | π/ 4 � • M Z QEC11: Second Interna1onal Conference on Quantum Error Correc1on 7
Infinite code family Approach 1: Concatenate a finite (stabilizer) quantum error‐correc1ng code FTQCP possible for any stabilizer code as a base, but CSS codes are par1cularly nice. • Nomenclature: “Level ” of concatena1on is an ‐qubit block. (Level 0 is “physical.”) n ℓ • ℓ Error‐detec:ng codes: lower levels locate errors for upper levels, so error‐detec1ng codes can correct them. • Numerous codes studied in the literature: • [[2, 1, 1]] Bell‐state (for LOQC) • [[5, 1, 3]] Five‐qubit • [[6, 2, 2]] C 6 • [[7, 1, 3]], [[49, 1, 9]] Steane/color (and concatenated Steane) • [[4, 1, 2]] (“ C 4 ”) [[9, 1, 3]], [[25, 1, 5]], [[49, 1, 7]], [[81, 1, 9]] Bacon‐Shor • [[15, 1, 3]] Reed‐Muller • [[13, 1, 3]], [[41, 1, 5]], [[85, 1, 7]] Surface • [[21, 3, 5]], [[60, 4, 10]] Reed‐Simon (“polynomial”) • [[21, 3, 5]], [[23, 1, 7]] Golay • [[43, 5, 7]], [[45, 3, 9]], [[47, 1, 11]], [[75, 5, 11]], [[77, 3, 13]], [[79, 1, 15]], [[97, 7, 13]], [[99, 5, 15]], [[101, • 3, 17]], [[103, 1, 19]] Quadra1c residue [[31, 11, 5]], [[63, 27, 7]], [[63, 39, 5]], [[79, 1, 15]], [[89, 1, 17]], [[103, 1, 19]], [[127, 1, 19]], [[127, 29, 15]], • [[127, 43, 13]] BCH QEC11: Second Interna1onal Conference on Quantum Error Correc1on 8
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