Fermions in Quantum Complexity Theory

Transcription

1 Fermions in Quantum Complexity Theory Edwin Ng MIT Department of Physics December 14, 2012

2 Occupation Number Formalism Consider n identical particles in m modes. If there are x j particles in the jth mode, the state is x 1,..., x m, where m j=1 x j = n.

3 Occupation Number Formalism Consider n identical particles in m modes. If there are x j particles in the jth mode, the state is x 1,..., x m, where m j=1 x j = n. States of this form (Fock states) make up a subspace H n, of states describing n particles in m modes. The full Fock space is given by H = H 1 H 2 For fermions, the sum ends at n = m, and dim H n = ( m n).

4 Occupation Number Formalism Consider n identical particles in m modes. If there are x j particles in the jth mode, the state is x 1,..., x m, where m j=1 x j = n. States of this form (Fock states) make up a subspace H n, of states describing n particles in m modes. The full Fock space is given by H = H 1 H 2 For fermions, the sum ends at n = m, and dim H n = ( m n). To go back to H = H n, (anti-)symmetrize over all possible combinations giving x j particles in mode j.

5 Moving Between Subspaces Start with the vacuum state 0,..., 0. To add a particle to mode j, apply the creation operator a j. By Hermitian adjoint, a j removes a particle from mode j (destruction operator).

6 Moving Between Subspaces Start with the vacuum state 0,..., 0. To add a particle to mode j, apply the creation operator a j. By Hermitian adjoint, a j removes a particle from mode j (destruction operator). For fermions, {a j, a k } = δ jk and {a j, a k } = {a j, a k } = 0. (a j )2 = 0 gives Pauli exclusion principle and implies x j {0, 1} for any nonzero state x 1,..., x m.

7 Moving Between Subspaces Start with the vacuum state 0,..., 0. To add a particle to mode j, apply the creation operator a j. By Hermitian adjoint, a j removes a particle from mode j (destruction operator). For fermions, {a j, a k } = δ jk and {a j, a k } = {a j, a k } = 0. (a j )2 = 0 gives Pauli exclusion principle and implies x j {0, 1} for any nonzero state x 1,..., x m. A state of n fermions in modes j 1,..., j n is created by ɛ j1 j n a j 1 a j n 0,..., 0 where ɛ j1 j n enforces sign convention.

8 Moving Between Subspaces Start with the vacuum state 0,..., 0. To add a particle to mode j, apply the creation operator a j. By Hermitian adjoint, a j removes a particle from mode j (destruction operator). For fermions, {a j, a k } = δ jk and {a j, a k } = {a j, a k } = 0. (a j )2 = 0 gives Pauli exclusion principle and implies x j {0, 1} for any nonzero state x 1,..., x m. A state of n fermions in modes j 1,..., j n is created by ɛ j1 j n a j 1 a j n 0,..., 0 where ɛ j1 j n enforces sign convention. The number of particles in mode j is the eigenvalue of N j = a j a j.

9 Fermionic Quantum Computation Any unitary operator acting on H can be generated by the set of creation and destruction operators: U = exp(ih), where H(a 1, a 1,..., a m, a m) is Hermitian.

10 Fermionic Quantum Computation Any unitary operator acting on H can be generated by the set of creation and destruction operators: U = exp(ih), where H(a 1, a 1,..., a m, a m) is Hermitian. If every term of H is quadratic, then operator involves noninteracting fermions.

11 Fermionic Quantum Computation Any unitary operator acting on H can be generated by the set of creation and destruction operators: U = exp(ih), where H(a 1, a 1,..., a m, a m) is Hermitian. If every term of H is quadratic, then operator involves noninteracting fermions. H must have an even number of a j and a j per term to give physical operators on fermions (preserves parity of fermions).

12 Fermionic Quantum Computation Any unitary operator acting on H can be generated by the set of creation and destruction operators: U = exp(ih), where H(a 1, a 1,..., a m, a m) is Hermitian. If every term of H is quadratic, then operator involves noninteracting fermions. H must have an even number of a j and a j per term to give physical operators on fermions (preserves parity of fermions). H must have the same number of a j and a j per term to preserve the number of particles.

13 Fermionic Quantum Computation Any unitary operator acting on H can be generated by the set of creation and destruction operators: U = exp(ih), where H(a 1, a 1,..., a m, a m) is Hermitian. If every term of H is quadratic, then operator involves noninteracting fermions. H must have an even number of a j and a j per term to give physical operators on fermions (preserves parity of fermions). H must have the same number of a j and a j per term to preserve the number of particles. Terhal and DiVincenzo (2002) showed quantum computation using a system of noninteracting fermions can be simulated classically, even with adaptive von Neumann measurements. Results were equivalent to a subclass of Valiant s matchgates (2001).

14 Noninteracting Fermions: Complexity Consider special case where number of fermions is preserved. A computation step acting between modes α and β is a unitary U = exp(ih) generated by H = h αα a αa β + h ββ a β a β + h αβ a αa β + h αβ a β a α V = exp(ih) is the single-particle unitary.

15 Noninteracting Fermions: Complexity Consider special case where number of fermions is preserved. A computation step acting between modes α and β is a unitary U = exp(ih) generated by H = h αα a αa β + h ββ a β a β + h αβ a αa β + hαβ a β a α V = exp(ih) is the single-particle unitary. Suppose U acts on a non-vacuum state with one fermion: Ua j 0 = Ua j U U 0 = Ua j U 0 where Ua j U = k V jk a k

16 Noninteracting Fermions: Complexity Now calculate y U x for x = a j 1 a j l 0 (j 1 < < j l ) and y = a k 1 a k l 0 (k 1 < k l ).

17 Noninteracting Fermions: Complexity Now calculate y U x for x = a j 1 a j l 0 (j 1 < < j l ) and y = a k 1 a k l 0 (k 1 < k l ). U x = Ua j 1 U Ua j k U 0 = ( ) V p 1 j 1 V p l j l a p 1 a p l 0 p 1,...,p l

18 Noninteracting Fermions: Complexity Now calculate y U x for x = a j 1 a j l 0 (j 1 < < j l ) and y = a k 1 a k l 0 (k 1 < k l ). so that y U x = U x = Ua j 1 U Ua j k U 0 = ( ) V p 1 j 1 V p l j l a p 1 a p l 0 p 1,...,p l p 1,...,p l ( ) V p 1 j 1 V p l j l 0 a kl a k1 a p 1 a p l 0

19 Noninteracting Fermions: Complexity Now calculate y U x for x = a j 1 a j l 0 (j 1 < < j l ) and y = a k 1 a k l 0 (k 1 < k l ). so that y U x = U x = Ua j 1 U Ua j k U 0 = ( ) V p 1 j 1 V p l j l a p 1 a p l 0 p 1,...,p l p 1,...,p l ( ) V p 1 j 1 V p l j l 0 a kl a k1 a p 1 a p l 0 The braket is zero unless p 1,... p l is a permutation of k 1,..., k l, and sgn(σ) if the permutation is σ.

20 Noninteracting Fermions: Complexity Rewrite sum in terms of permutations: y U x = sgn(σ)v σ(k 1) j 1 V σ(k l ) j l = det Ṽ σ S l where Ṽ is l l a submatrix of V with selected rows j 1,..., j l and columns k 1,..., k l.

21 Noninteracting Fermions: Complexity Rewrite sum in terms of permutations: y U x = sgn(σ)v σ(k 1) j 1 V σ(k l ) j l = det Ṽ σ S l where Ṽ is l l a submatrix of V with selected rows j 1,..., j l and columns k 1,..., k l. This is the expression claimed in lecture. The probability distribution is det Ṽ 2, and can be sampled efficiently in classical polynomial time.

22 Noninteracting Fermions: Complexity Rewrite sum in terms of permutations: y U x = sgn(σ)v σ(k 1) j 1 V σ(k l ) j l = det Ṽ σ S l where Ṽ is l l a submatrix of V with selected rows j 1,..., j l and columns k 1,..., k l. This is the expression claimed in lecture. The probability distribution is det Ṽ 2, and can be sampled efficiently in classical polynomial time. Quantum computation using noninteracting fermions and preserving the number of particles can be efficiently simulated classically.

23 Noninteracting Fermions: General Cases More generally, quantum computation involving noninteracting fermions need not conserve particle number, as long as it conserves particle parity. The more general case uses creation and destruction operators c 2j = a j + a j and c 2j+1 = i(a j a j ), satisfying {c k, c l } = 2δ kl. Terhal and DiVincenzo proved this can also be classically efficiently simulated.

24 Noninteracting Fermions: General Cases More generally, quantum computation involving noninteracting fermions need not conserve particle number, as long as it conserves particle parity. The more general case uses creation and destruction operators c 2j = a j + a j and c 2j+1 = i(a j a j ), satisfying {c k, c l } = 2δ kl. Terhal and DiVincenzo proved this can also be classically efficiently simulated. Furthermore, complete von Neumann measurements also are not sufficient to bring the sysem to full universality, but it is for bosons KLM (2007)

25 Universal Fermionic Quantum Computation A universal set for quantum computation using interacting fermions was given by Bravyi and Kitaev (2000) for local fermion modes, consisting of the following unitaries: ( ) exp i π 4 a 0 a 0 (potential term) ( ) exp i π 4 (a 0 a 1 + a 1 a 0) (tunnelling term) ) (i π 4 (a 1a 0 + a 0 a 1 ) exp exp (superconductor interaction term) ( ) iπa 0 a 0a 1 a 1 (two-particle interaction term)

26 Universal Fermionic Quantum Computation A universal set for quantum computation using interacting fermions was given by Bravyi and Kitaev (2000) for local fermion modes, consisting of the following unitaries: ( ) exp i π 4 a 0 a 0 (potential term) ( ) exp i π 4 (a 0 a 1 + a 1 a 0) (tunnelling term) ) (i π 4 (a 1a 0 + a 0 a 1 ) exp exp (superconductor interaction term) ( ) iπa 0 a 0a 1 a 1 (two-particle interaction term) Beenakker, DiVincenzo, Emary, and Kindermann in 2004 showed that allowing for charge detection in noninteracting fermions gives full quantum computation. Constucted a CNOT using beamsplitters, charge detectors, and one ancilla.

27 References S.B. Bravyi and A.Y. Kitaev. Fermionic quantum computation. quant-ph/ (2000). B.M. Terhal and D.P. DiVincenzo. Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, (2002). quant-ph/ C.W.J. Beenakker, D.P. DiVincenzo, C. Emary, M. Kindermann. Charge detection enables free-electron quantum computation. Phys. Rev. Lett. 93, (2004). quant-ph/