� ✠ ✂ � ✞ � ✟ ✠ ✂ ✟ � ✂ � ✂ Classical v quantum In a classical computer, each bit of information is stored by a transistor containing trillions of electrons. Quantum Error Correction On a quantum computer, a single electron or nucleus Peter J. Cameron in a magnetic field carries a bit of information. Interaction with the environment is much more School of Mathematical Sciences serious. Queen Mary and Westfield College London E1 4NS, U.K. Decoherence puts a limit on the space and time p.j.cameron@qmw.ac.uk resources available to a quantum computer. MathFIT London, 5 April 2000 In order to get round this limit, the computer must be fault tolerant , that is, it must have error correction built in; and the error correction circuits should not introduce more errors than they correct! 1 3 Classical error correction ✁ 2 ✄ 0 GF ☎ 1 ✆ . An element of F is a bit of Why quantum computing? Let F information. A word of length n (an element of F n ) contains n bits of information. V In 1990 Peter Shor proved the following theorem. A code is a subset C of V such that any two elements Theorem 1 There exists a randomized algorithm for of C are far apart. We only use codewords to carry integer factorization which runs in polynomial time on information; if few errors occur, the correct codeword a quantum computer. is likely to be the nearest. ✁ v ☎ w On a classical computer, primality testing is ‘easy’ but ✝ V , the Hamming distance d For v ☎ w ✂ is the factorization is ‘hard’. This is the basis of the RSA number of coordinates i such that v i w i . cryptosystem. If the minimum Hamming distance between distinct Roughly speaking, a quantum computer is highly elements of C is d , then C can correct up to ✁ d parallel; we can run exponentially many 1 ✡ 2 ☛ errors. So an error pattern is correctable ✁ d computations at the same time, and only those which 1 ✡ 2 if it has weight at most ☛ . terminate with a positive result will produce output. ✁ v ✁ v ☎ 0 The weight of v is wt d ✂ . If C is linear, then its minimum distance is equal to its minimum weight. 2 4
✔✕ ✌ � � ✓ ✁ ✌ ✔✕ ✑ ✑ ✒ ✒ ✒ ✑ ☎ ✌ ✔✕ � ✒ ✓ ✂ ✔✕ ✔✕ ✔✕ ✔✕ ✔✕ � ✓ ✒ ✓ ✓ ☎ � ☎ ✑ ✒ ✁ � ✂ ✂ ✓ � ✦ ✜ ✂ ✁ � ✥ ✌ ✂ ✤ ✤ ✧ ✁ ✙ � ✍ ✂ ✁ ✂ ✎ ✄ � ✏ ✂ � Quantum errors States and observables An error, like any physical process, is a unitary The state of a quantum system is a unit vector in a transformation of the state space. The space of complex Hilbert space. An observable is a errors to a single qubit is 4 -dimensional, and is self-adjoint operator on the state space, whose spanned by the four unitary matrices eigenvalues are the possible values of the I e 0 e 0 , e 1 e 1 (no error) observable. X e 0 e 1 , e 1 e 0 (bit error) Z e 0 e 0 , e 1 ✠ e 1 (phase error) The interpretation of the coefficients a i of a state Y iXZ (combination) vector with respect to an orthonormal basis of ☞ 2 is the ☎ Z are the Pauli spin matrices . ☞ a i Note that I ☎ X ☎ Y eigenvectors of an observable is that probability of obtaining the corresponding eigenvalue ✂ v e v . ✠ 1 We can write Xe v e v ✖ 1 , Ze v as the value of a measurement. 5 7 Bits and qubits Quantum errors The quantum analogue of a bit of information is called a qubit . It is the state of a system in a 2 spanned by e 0 and 2 -dimensional Hilbert space Now the errors to n qubits act coordinatewise, and ✁ a ✁ b e 1 , where e 0 and e 1 are eigenvectors corresponding are generated by X ✂ and Z ✂ for a ☎ b ✝ V , where to the eigenvalues 0 and 1 of the qubit. ✁ a ✁ b ✂ v ✗ b e v ✂ : e v ✂ : e v ✠ 1 X e v Z ✖ a Thus, the qubit is represented by the self-adjoint These generate the error group , an extraspecial matrix ✁ E ✘ I . 2 -group E of order 2 2 n ✖ 1 with centre Z 0 0 0 1 ✁ 2 ✁ E ✂ 2 n ; we represent the coset GF E E ✡ Z relative to this basis. So in the state α e 0 β e 1 , the ✁ a ✁ b ✁ a ✘ X ☞ 2 and ☞ 2 probabilities of measuring 0 and 1 are ☞ α ☞ β ✂ Z ☞ b ✆ by ✂ . respectively. On E , we have a quadratic form q given by An n -tuple of qubits is an element of the tensor ✁ X ✁ a ✁ b ✜ I ✂ 2 ✂ q ✚ a ✛ b ✠ 1 product ✂ Z 2 2 2 n and associated symplectic form ✢ given by ✣ X ✁ a ✁ b ✁ a ✁ b a basis for this space consists of all vectors ✜ I ✚ a ✛ b ✚ a ✧ ✛ b ✠ 1 ✂ Z ☎ X ✂ Z e v e v 1 e v n ✁ v 1 for v ☎ v n ✝ V . 6 8
✝ ✏ ✩ ☎ ✪ ☞ � ✏ ✂ ✠ ✂ ✝ ✪ ✟ ✠ ✡ ✂ ✔✕ ✂ ✩ ✟ ✠ ✓ ✂ ✆ ☛ ✂ ✏ ✝ � ✪ ✌ ✫ ★ ✪ ✩ ✝ ★ ★ Quantum codes Let S be an abelian subgroup of E such that S is GF ✮ 4 ✯ to quantum totally singular (w.r.t. q ). Then under the action of S , 2 n is the sum of ✁ 4 ☞ S the state space ☞ orthogonal The field GF ✂ can be written as eigenspaces. Let Q be an eigenspace. Then ✁ 2 ✄ a ω b ω : a GF ☎ b ✁ 4 the error group permutes the eigenspaces ✂ n , So we have a bijection θ between E and GF ✁ a regularly; a ω b ω . ☞ b given by ✁ 4 the stabilizer of Q is S ✩ ; ✂ n is totally isotropic Moreover, if a subspace of GF with respect to the Hermitian inner product on ✁ 4 S acts trivially on Q . ✂ n , then its image in E is totally singular. GF ✁ a Thus, errors in S are undetectable, while errors in S ☞ b Also, the quantum weight of ✂ is equal to the is a subset of E with the Hamming weight of a ω b ω . have no effect. So if property ✁ 4 ✬ 1 e e ☎ f f S ✭ S So good GF ✂ -codes can be used to construct good quantum codes. then errors in can be corrected. (If two such errors have undetectably different effect, then they have the same effect!) 9 11 Quantum error correction 2 r , then The subspace Q is our quantum code. If ☞ S ✁ Q dim 2 n ✬ r ; we can think of Q as consisting of n r qubits “smeared out” over the space of n qubits. ✁ a Define the quantum weight of ☞ b E to be the number of coordinates i such that either a i or b i (or both) is non-zero, that is, some error has occurred in the i th qubit. By taking to consist of all errors with quantum ✁ d 1 ✡ 2 weight at most ☛ , Calderbank, Rains, Shor and Sloane proved the following analogue of classical error correction: Theorem 2 Suppose that the minimum quantum ✁ d weight of S ✭ S is d . Then Q corrects 1 ✡ 2 qubit errors. 10
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