Quantum Error Correction ISSQUI 05 International Summer School on Quantum Information MPIPKS, Dresden, August 29 – September 30, 2005 Quantum Algorithms & Quantum Error Correction Martin Rotteler¨ & Markus Grassl Arbeitsgruppe Quantum Computing, Prof. Beth Institut f¨ur Algorithmen und Kognitive Systeme Fakultat¨ f¨ur Informatik, Universitat¨ Karlsruhe (TH) Germany http://iaks-www.ira.uka.de/QIV Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 QECCs: The Basic Problem storage of quantum information transmission quantum state =⇒ channel =⇒ erroneous state ~ w errorsw Errors are induced by, e. g., interaction with an environment, coupling to a bath, or also by imperfect operations. Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Error Correction General scheme environment no access s system |φi inter- action entanglements ⇒ decoherence Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Error Correction General scheme environment no access s system |φi s s encoding ancilla |0i interaction encoding s three-partys entanglement Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Error Correction General scheme environment no access s s system |φi s s s code entanglement encoding ancilla |0i interaction encoding s s s syndrome ancilla |0i error correction s environment/ancilla Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Error Correction General scheme environment no access s s s system |φi |φi s s s encoding ancilla |0i |0i interaction decoding encoding s s s syndrome ancilla |0i error correction s s Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Error Correction General scheme environment no access s s s system |φi |φi s s s encoding ancilla |0i |0i interaction decoding encoding s s s syndrome ancilla |0i error correction s s Basic requirement some knowledge about the interaction between system and environment Common assumptions • no initial entanglement between system and environment • local or uncorrelated errors, i. e., only a few qubits are disturbed =⇒ CSS codes, stabilizer codes • interaction with symmetry =⇒ decoherence free subspaces Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Interaction System/Environment “Closed” System environment |εi = Uenv/sys |εi|φi system |φi inter- action ) “Channel” Q: ρin := |φihφ| 7−→ ρout := Q(|φihφ|) := EiρinEi† i X with Kraus operators (error operators) Ei Local/low correlated errors n • product channel Q⊗ where Q is “close” to identity • Q can be expressed (approximated) with error operators E˜i such that each Ei acts on few subsystems, e.g. quantum gates Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Computer Science Approach: Discretize QECC Characterization [Knill & Laflamme, PRA 55, 900–911 (1997)] A subspace C of H with orthonormal basis {|c1i ,..., |cK i} is an error-correcting code for the error operators E = {E1, E2,...}, if there exists constants αk,l ∈ C such that for all |cii, |cji and for all Ek, El ∈E: hci| Ek†El |cji = δi,jαk,l. (1) It is sufficient that (1) holds for a vector space basis of E. Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Linearity of Conditions Assume that C can correct the errors E = {E1, E2,...}. New error-operators: A := λkEk and B := µlEl k l X X hci| A†B |cji = λkµl hci| Ek†El |cji k,l X = λkµlδi,jαk,l k,l X = δi,j · α0(A, B) It is sufficient to correct error operators that form a basis of the linear vector space spanned by the operators E. =⇒ only a finite set of errors. Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Error Basis Pauli Matrices 0 1 0 −i 1 0 1 0 σx := , σy := , σz := , I := 1 0! i 0 ! 0 −1! 0 1! • vector space basis of all 2 × 2 matrices • unitary matrices which generate a finite group Error Basis for many Qubits/Qudits E error basis for subsystem of dimension d with I ∈E n =⇒E⊗ error basis with elements E := E1 ⊗ ... ⊗ En, Ei ∈E weight of E: number of factors Ei 6= I Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Local Error Model Code Parameters C = [[n,k,d]] n: number of subsystems used in total k: number of (logical) subsystems encoded d: “minimum distance” – correct all errors acting on at most (d − 1)/2 subsystems – detect all errors acting on less than d subsystems – correct all errors acting on less than d subsystems at known positions Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Quantum Errors Bit-flip error: • Interchanges |0i and |1i. Corresponds to “classical” bit error. £ ¡ 0 1 ¤ Given by NOT gate X ¡ ¤ • = ¢ ¥ 1 0 Phase-flip error: • Inverts the relative phase of |0i and |1i. Has no classical analogue! £ ¡ 1 0 ¤ Given by the matrix Z ¡ ¤ • = ¢ ¥ 0 −1 Combination: £ ¤ ¡ 0 −1 ¤ Combining bit-flip and phase-flip gives Y ¡ XZ. ¥ • = ¢ = 1 0 Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Pauli and Hadamard matrices “Pauli” matrices: ¢ ¢ ¢ 0 1 1 0 0 −1 I, X = ¡ £ , Z = ¡ £ , Y = XZ = ¡ £ 1 0 0 −1 1 0 ¢ 1 1 1 ¡ £ Hadamard matrix: H = √2 1 −1 Important properties: • H†XH = Z, “H changes bit-flips to phase-flips” ¢ 0 1 • ZX = ¡ £ = −Y = −XZ, “X and Z anticommute” −1 0 • All errors either commute or anticommute! Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Discretization of Quantum Errors Consider errors E = E1 ⊗ ... ⊗ En, Ei ∈ {I,X,Y,Z}. Pauli matrices: 0 1 1 0 0 −1 I, X = , Z = , Y = XZ = 1 0 0 −1 1 0 The weight of E is the number of Ei 6= 1. E.g., the weight of I ⊗ X ⊗ Z ⊗ Z ⊗ I ⊗ Y ⊗ Z is 5. Theorem: (later) If a code C corrects errors E of weight t or less, then C can correct arbitrary errors affecting ≤ t qubits. Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Repetition Code classical: sender: repeats the information, e.g. 0 7→ 000, 1 7→ 111 receiver: compares received bits and makes majority decision quantum mechanical “solution”: sender: copies the information, e.g. |ψi = α |0i + β |1i 7→ |ψi|ψi|ψi receiver: compares and makes majority decision but: unknown quantum states can neither be copied nor can they be disturbance-free compared Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 The No-Cloning Theorem Theorem: Unknown quantum states cannot be copied. Proof: The copier would map |0i |ψini 7→ |0i |0i, |1i |ψini 7→ |1i |1i, and (α |0i + β |1i) |ψini 7→ α |0i |0i + β |1i |1i 6= (α |0i + β |1i) ⊗ (α |0i + β |1i) 2 2 = α |0i |0i + β |1i |1i + αβ(|0i |1i + |1i |0i) Contradiction to the linearity of quantum mechanics! Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 The Basic Idea Classical codes Quantum codes Partition of the set of all words of length Orthogonal decomposition of the n 2 n over an alphabet of size 2. vector space H⊗ , where H=∼C . • ¡ ••• • E n k ¡ 2 - 1 • • ••• ¡ ¡ − ••• • • • ¡ ¡ • • • ¡ ¡ • • • ¢ • • ••• n E • • ••• • • C2 i • ¡ • • • ¡ • ¡ ¡ ••• • • ¡ E • ¡ 1 • ••• ¡ ¡ • £ C • codewords n H⊗ = C ⊕ E1 ⊕ ... ⊕ E2n−k 1 • errors of bounded weight − Encoding: |xi 7→ Uenc(|xi|0i) • other errors Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Simple Quantum Error-Correcting Code Repetition code: |0i 7→ |000i, |1i 7→ |111i Encoding of one qubit: α |0i + β |1i 7→ α |000i + β |111i . This defines a two-dimensional subspace H ≤H2 ⊗H2 ⊗H2 C bit-flip quantum state subspace no error α |000i + β |111i (1 ⊗ 1 ⊗ 1)H C 1st position α |100i + β |011i (X ⊗ 1 ⊗ 1)H C 2nd position α |010i + β |101i (1 ⊗ X ⊗ 1)H C 3rd position α |001i + β |110i (1 ⊗ 1 ⊗ X)H C Hence we have an orthogonal decomposition of H2 ⊗H2 ⊗H2 Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Simple Quantum Error-Correcting Code Problem: What about phase-errors? Phase-flip Z: |0i 7→ |0i and |1i 7→ − |1i. In the Hadamard basis |+i , |−i given by 1 |+i = √2 (|0i + |1i), 1 |−i = √2 (|0i − |1i) the phase-flip operates like the bit-flip Z |+i = |−i, Z |−i = |+i. To correct phase errors we use repetition code and Hadamard basis: 1 1 |0i 7→ (H ⊗ H ⊗ H) √2 (|000i + |111i)= 2 (|000i + |011i + |101i + |110i) 1 1 |1i 7→ (H ⊗ H ⊗ H) √2 (|000i − |111i)= 2 (|001i + |010i + |100i + |111i) Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Simple Quantum Error-Correcting Code phase-flip quantum state subspace α no error 2 (|000i + |011i + |101i + |110i) (1 ⊗ 1 ⊗ 1)H C β + 2 (|001i + |010i + |100i + |111i) st α − − 1 position 2 (|000i + |011i |101i |110i) (Z ⊗ 1 ⊗ 1)H C β − − + 2 (|001i + |010i |100i |111i) nd α − − 2 position 2 (|000i |011i + |101i |110i) (1 ⊗ Z ⊗ 1)H C β − − + 2 (|001i |010i + |100i |111i) rd α − − 3 position 2 (|000i |011i |101i + |110i) (1 ⊗ 1 ⊗ Z)H C − β − 2 (|001i + |010i + |100i |111i) We again obtain an orthogonal decomposition of H2 ⊗H2 ⊗H2 Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Simple Quantum Error-Correcting Code |ψi |c1i s s s |0i |c2i g s |0i |c3i g s s |0i ¨¢¢ s1 g g q |0i ¨AK s2 g g q encoding syndrome computation measurement Error X ⊗ I ⊗ I gives syndrome s1s2 = 10 Error I ⊗ X ⊗ I gives syndrome s1s2 = 01 Error I ⊗ I ⊗ X gives syndrome s1s2 = 11 Markus Grassl 19. September 2005 Quantum Error Correction ISSQUI 05 Simple Quantum Error-Correcting Code |ψi |c1i s s s g |0i |c2i g s g |0i |c3i g s s g |0i |s1i g g s c s |0i |s2i g g c s s encoding syndrome computation error correction • Coherent error correction by conditional unitary transformation.