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4. A universal gate is the Deutsch gate which depends on an angular parameter θ
H 3 →
{
H 3
i cos θ(a, b, c) + sin θ(a, b, 1 − c) if a = b = 1
(a, b, c) ↦→
(a, b, c) else
For θ = π , we recover the classical Toffoli gate. This is taken as an argument that all
2
operations possible in classical computing are possible in quantum computing.
5.5.5 Quantum codes
We can now define quantum codes. For a general reference, see [FKLW].
Definition 5.5.11
Denote by H again the complex group algebra of Z 2 . We call a tensor product V := H ⊗n a
discrete quantum medium. (Think of a system composed of n spin 1/2 particles.)
1. A quantum code is a linear subspace W ⊂ V of a quantum medium V . Sometimes, a
quantum code is also called a protected space.
2. Let 0 ≤ k ≤ n. A k-local operator is a linear map O : V → V which is the identity on
n − k tensorands of Y . (Quantum gates are thus at least 2-local.)
3. Denote by π W : V → W the orthogonal projection. A quantum code W ⊂ V is called a
k-code, if the linear operator
π W ◦ O : W → W
is multiplication by a scalar for any k-local operator O.
One can show the following analogue of a lemma 5.5.5:
Lemma 5.5.12.
If W is a k-code, then information cannot be degraded from errors operating on less than k 2 of
the n particles.
Remarks 5.5.13.
1. A first attempt to realize qubits might be to take isolated trapped particles, individual
atoms, trapped ions or quantum dots. Such a configuration is fragile and one has to
minimize any external interaction. On the other hand, external interaction is need to
write and read off information.
The idea of topological quantum computing is to use non-local degrees of freedom to
produce fault tolerant subspaces. Concretely, one needs non-abelian anyons in quasi twodimensional
systems.
2. Storage devices are typically effectively two-dimensional. Thus the complex vector space
of qubits should be the space of states of a three-dimensional topological field theory. Maps
describing gates and circuits are obtained from colored cobordisms, i.e. three-manifolds
containing links. For example, the quantum analogue of the XOR gate, the CNOT gate
can be realized to arbitrary precision by braids.
3. A theorem of Freedman, Kitaev and Wang asserts that quantum computers and classical
computers can perform exactly the same computations. But their efficiency is different,
e.g. for problems like factoring integers into primes.
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