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|1| Quantum Error Correction: Optimizing Implementations of the [7,1,3] Code Yaakov S. Weinstein MITRE Quantum Information Science Group April 23, 2014 |2| What Can a Quantum Computer Do? Solve mathematical problems intractable on classical computers! Break RSA encryption codes! Classical Cryptography based on the RSA code is secure provided an eavesdropper cannot factor large numbers. Shor’s algorithm allows a quantum computer to factor large numbers in a short time! Typical Quantum Mechanic with his quantum computer Classical Channel Alice RSA encrypted line Bob |3| What Else Can a Quantum Computer Do? Faster database searches (i.e. pattern matching)! Ann’s Bob’s Charlie’s CLASSICAL ORACLE 0 or 1 0 or 1 0 or 1 Nth person Classically ~ N/2 lookups Ann’s Bob’s Charlie’s QUANTUM ORACLE 1 0 1 12 12 2 0 1 0 1 Nth person Grover’s algorithm for a quantum computer ~ N lookups Other algorithms have been discovered that allow a quantum computer to perform better than a classical computer. |4| How Does a Quantum Computer Work? Bits vs. Qubits Bit stored in capacitor “0 or 1” Qubit stored in spin ½ particle phase - + • Two possible states: charged or unchaged • Millions of electrons determine charge state 𝟏 (|𝟎 𝟐 𝟏 (|𝟎 𝟐 |𝟎 -|𝟏 ) 𝟏 (|𝟎 𝟐 +i|𝟏 ) -i|𝟏 ) 𝟏 (|𝟎 𝟐 +|𝟏 ) |𝟏 • Qubit can be in a continuum of states • Most states are superpositions of 0 and 1 Measuring a qubit ‘collapses’ the state to “0 or 1” |5| Superposition One qubit can be in 2 states at the same time. Two qubits can be in 4 states at the same time. n qubits can be in 2n states at the same time! So… On a ‘classical’ computer one can apply an algorithm to one input at a time On a quantum computer one can put the input into a superposition of a huge number of states and then apply the algorithm to all inputs simultaneously! But… The measurement outcome is quantum probabilistic. So even though the algorithm is applied to many states you can only measure one answer and which one you get is probabilistic! |6| Entanglement Entanglement is even more strange! Measur e A B A B A ½ the time B All of the time A ½ the time B All of the time •Superdense Coding - send two classical bits of information via one qubit •Quantum Cryptography - guaranteed secret by the laws of physics •Quantum Teleportation - send quantum information without sending a physical qubit •Algorithms - Shor’s factoring algorithm, Grover’s search algorithm, Others |7| Quantum Circuit Paradigm The gate paradigm is similar to classical computer architecture. Distinct gates are sequentially applied to physically recognizable qubits. Unitary operators: Two-qubit gates 1 2 3 qubits One-qubit gates time |8| So Why Don’t We Have a Quantum Computer? Decoherence – the interaction between a quantum systems and its environment causing perceived non-unitary evolution when looking at the system dynamics alone. Decoherence can cause conventional (bit flips) and quantum (dephasing) errors and tends to destroy superpositions… Error Correcting – monitor system via syndrome measurements for feedback control Many known codes [7,1,3], [5,1,3], surface codes, etc. Error Avoidance – Bang-bang control – average out environment effects via open loop control techniques Decoherence Free Subspace (DFS)– encode information into system subspace Noiseless Subsystem (NS) – encode information into a system degree of freedom |9| 𝛼|0 + 𝛽|1 |0 |0 |0 |0 |0 |0 ENCODE Quantum Error Correction 0𝐿 = |0000000 + |0001111 + |0110011 + |0111100 + |1010101 + |1011010 +1 |1100110 + |1101001 𝐿 Logical Basis States = |1111111 + |1110000 + |1001100 + |1000011 + |0101010 + |0100101 0011001code-words, + |0010110 Distance+3 |between one error can be corrected An [n,k,d] QEC code will encode k qubits of quantum information in n physical qubits with distance d between code-words | 10 | Quantum Fault Tolerance (QFT) Allows for successful computation despite errors in basic components This can be done by concatenating quantum error correcting codes QEC Level 1 QEC Level 1 QEC QEC QEC QEC Level 1 QEC QEC QEC Level 1 QEC QEC QEC Level 1 QEC QEC QEC Level 1 QEC QEC QEC Level 1 QEC QEC Level 1 QEC QEC Level 3 Level Level Level2 Level QEC 1 1 1 QEC Level 1 QEC Level 1 QEC Level Level Level2 Level QEC 1 1 1 Level 1 QEC Level 1 QEC QEC Level 1 QEC Level Level Level2 Level QEC 1 1 1 QEC Level 1 QEC Level 1 QEC Level Level Level2 Level QEC 1 1 1 QEC Level 1 QEC Level 1 QEC Level Level Level2 Level QEC 1 1 1 QEC Level 1 QEC Level 1 Level Level Level2 Level QEC 1 1 1 QEC QEC Level 1 QEC Level 1 QEC Level 1 QEC Level 1 QEC Level 1 QEC Level 1 QEC Level 1 Then, if you keep your error probability on basic components below a calculated threshold, you can successfully compute despite errors QEC Level 1 QEC Level 1 QEC Level 1 QEC QEC QEC Level Level Level2 Level QEC 1 1 1 Single Useful rules for making your computation fault tolerant: qubit Single qubit encoded 1. Keep your information in the error correcting codeinto manifold 7 physical qubits QEC QEC QEC QEC QEC QEC QEC • Or else errors will occur when you’re outside of the correcting Level 1 Level 1 QEC Level 1 Level 1 Level2 1 Level 1 Levelerror 1 Level will protect against all space one qubit errors 2. Do not reuse qubits Now have 49 qubits and can protect 3. Do not allow a qubit to interact with more than one other qubit against two singlequbit errors 4. Repeat to ensure no errors have occurred | 11 | Violations of QFT Rules: Syndrome Measurements How do we construct these? (no Hadamards) Shor State (no Hadamards) Shor State (no Hadamards) Shor State | 12 | Violations of QFT Rules: Encoding | 13 | Error Model Non-equiprobable Pauli error environment on single qubit: 𝜌𝑓 = 1 − 𝑝𝑥 − 𝑝𝑦 − 𝑝𝑧 𝜌𝑖 + 𝑝𝑥𝜎𝑥𝜌𝑖𝜎𝑥 + 𝑝𝑦𝜎𝑦𝜌𝑖𝜎𝑦 + 𝑝𝑧 𝜎𝑧𝜌𝑖𝜎𝑧 𝑝0 We assume that any qubit taking part in a quantum gate (including initialization and measurement) undergoes no errors with probability p0, an error sx with probability px, sy with probability py, sz with probability pz. Where sj are the Pauli spin matrices and s0 is the identity matrix. CNOT gate errors: p0, px, py, pz 0,𝑥,𝑦,𝑧 errors: p0, px, py, pz no errors 𝑝𝑎 𝑝𝑏 𝜎𝑎𝑗𝜎𝑏𝑘𝐶𝑗 𝑁𝑂𝑇𝑘 𝜌𝑖 𝐶𝑗 𝑁𝑂𝑇𝑘𝜎𝑎𝑗𝜎𝑏𝑘 𝜌𝑓 = 𝑎,𝑏 | 14 | Shor State Construction Parity Check GHZ ConstructionVerification |0 |0 |0 |0 A Shor state appropriate for the syndrome measurement of a [[7,1,3]] code is done by building a highly-entangled GHZ state. This state is verified to ensure multiple errors have not taken place during construction. Hadamard gates complete the Shor state construction. Y.S. Weinstein and S. D. Buchbinder, PRA 86, 052336 (2012); How many verifications steps should we use? Or is it even worth using Shor states in the first place – maybe stick with one-qubit ancillae? Logical Zero: 1 Qubit No verifications 1 verification 2 verifications 7-Qubit fidelity 1-25px28py-21pz 1-85px-37py-12pz 1-55px-19py12pz 1-55px-19py12pz 1-Qubit fidelity 1-9px-9py 1-25px-11py 1-19px-7py 1-19px-7py After perfect QEC 1-12px2 1-92px2-74pxpy-14py2 1 1 | 15 | Encoding a Qubit in a [7,1,3] Code (no Hadamards) Shor State (no Hadamards) Shor State (no Hadamards) Shor State But how do we encode our information without error? Two methods: 1) Apply a gate sequence that will properly encode the information we wish to preserve (and hope that the errors are not too bad) – not fault tolerant – errors can spread 2) Apply fault tolerant error correction to seven qubits all in the state zero state, projecting it into the code space S.D. Buchbinder, C.L. Huang, and Y.S. Weinstein, QIP 12, 699 (2013); Y. S. Weinstein, PRA 84, 012323 (2011). | 16 | Gate Fidelities 0 0 0 Perfect DECODE 0 Noisy GATE 0 Perfect ENCODE 0 7-Qubit Fidelity (fidelity of process) OR Perfect Error Correction 1-Qubit Fidelity (fidelity of protected information) After Perfect QEC Fidelity (correctability of errors) | 17 | Clifford Gates For codes such at the [[7,1,3]] code Clifford gates, C, such as the NOT-gate, Phase gate, Hadamard gate, and CNOT gate, can be implemented bit-wise: C† C† Clifford Gate 7-qubit fidelity After Perfect QEC C† Hadamard 1-7px-7py-7pz 1 NOT 1-3px-3py-3pz 1 Phase 1-7px-7py-7pz 1 C† C† C† C† One logical qubit fidelities are of the same order as the 7-qubit fidelities but have additional terms dependent on the initial encoded state. We can also determine single logical qubit cmatrices which allows us to calculate (state independent) gate fidelities. Y. S. Weinstein and S. D. Buchbinder, J. Mod. Opt., 61, 49 (2014) | 18 | Y.S. Weinstein, PRA 87, 032320 (2013) Non-Clifford T-Gate T-Gate Fault Tolerant Logical Zero Gate Sequence Logical Zero 7-Qubit Fidelity 1-7px-7py26pz 1-10px-13py-35pz ZPX c-Matrix Fidelity 1-3px-5py14pz 1-6px-11py-23pz 1 1 Noisy Encode Logical Zero ZPX ZPX ZPX After Perfect QEC ZPX T L ZPX X X MEASURE X2 MEASURE Noisy 7-Qubit Shor State Perfect Encoding of Arbitrary State ZPX X X X X X | 19 | How Often Quantum Error Correction? |0 |0 |0 DECODE |0 Recovery |0 GATE |0 ENCODE 𝛼|0 + 𝛽|1 Syndrome Measurement Apply N times Since all gates are imperfect the application of the syndrome measurements will also introduce errors The application of syndrome measurements is very expensive in terms of time and number of qubits | 20 | Quantum Error Correction During Many Gates |0 |0 |0 p p p p p p p p p p p p p p GATE |0 p GATE |0 p GATE |0 ENCODE 𝛼|0 + 𝛽|1 … p … p p p P(one error): 7p-O(p2) 14p-O(p2) 7np-O(p2) P(two errors): 21p2-O(p3) 84p2-O(p3) 21[n+2 p 𝑛 2 ] p2-O(p3) | 21 | Quantum Error Correction During 50 Gates How often should we apply QEC such that the output state of our computation is as accurate as possible? Better or worse to apply QEC after every gate? Infidelity Fractional change of Infidelity: 𝐷 𝐼50 , 𝐼𝑞 = Y. S. Weinstein, PRA 89, 020301(R) (2014); Y. S. Weinstein, PRA 88, 012325 (2013); Y. S. Weinstein, PRA 87, 032320 (2013). 𝐼50 − 𝐼𝑞 𝐼50 | 22 | Infidelity Evolution During 50 Gates Infidelity (to first order) vs. gate # is fit by line Infidelity decreases after T-gate X – 50 O – 20 □ – 10 ◊–2 ●–1 ▲– 0 Y. S. Weinstein, working on it... | 23 | Conclusions & Ongoing Work Quantum computers offer revolutionary improvement in computing power due to the ability of qubits to be in superposition states and to become entangled. Powerful quantum algorithms already known include Shor’s algorithm to factor large numbers and Grover’s fast database searches Careful explorations of the basic elements involved in implementing fault-tolerant quantum computation. I have demonstrated means of increasing the accuracy of these basic elements Analysis of how often QEC should be applied during a sequence of gates Ongoing work – continued exploration of QEC during multiple gates and especially the effect of T-gates Ongoing work – exploration of different types of syndrome measurements Future work – other codes including other concatenated codes and surface codes