Invariants of groups and universality of quantum gate sets
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1 Invariants of groups and universality of quantum gate sets Gábor Ivanyos MTA SZTAKI RWCA 2008, Levico Terme, June 19, 2008.
2 Reversible gates Universality of reversible gates Reversible gates Universality Jeandel s separation result reversible n-bit gates: bijections {0, 1} n {0, 1} n reversible circuits on K > n bits: sequences of reversible n-bit gates together with wirings to n of the K bits: injections {1,..., n} {1,..., K} equivalent: sequences of reversible gates on the first n bits and permutations of the K bits
3 Reversible gates Universality Jeandel s separation result Wired gates and an equivalent circuit using permutations A C B A B C (14253) (35241)*(12) (12)
4 Universality Universality of reversible gates Reversible gates Universality Jeandel s separation result Γ: a set of n-bit reversible gates K-universality: every π A 2 K can be realized by a circuit built from Γ: A 2 K is generated by Γ and the bit permutations Remark: For K > n, A 2 K is the largest possible... universality: K-universality for all K K 0. Monotone property: For L > K > 2 K-universality = L-universality, universality=k-universality form some K max{3, n}.
5 Known facts Universality of reversible gates Reversible gates Universality Jeandel s separation result Case K = 2: the 2-bit reversible gates are not universal: S 4 = AGL(2, 2): only affine maps can be realized De Vos, Storme 2004: Γ S 2 n is universal in a much weaker sense Γ AGL(2 n, 2) Toffoli gate: if x 1 = x 2 = 1 then flip x 3 3-universal together with the one-bit gate flip x 1
6 Jeandel s separation result Reversible gates Universality Jeandel s separation result For n = 2 r + 3, universal n-bit gate set which is not 2n 8-universal. Γ 0 : a 3-universal set involutive gates on 3 bits. (E.g., the transpositions in S 8.) The gates on r qubits { (γ(v), w) if w = (1,..., 1) or (0,..., 0), γ(v, w) = (v, w) otherwise. Γ = { γ γ Γ 0 }.
7 Separation 2. Universality of reversible gates Reversible gates Universality Jeandel s separation result On K = 2 r+1 2 bits, (0 K/2, 1 K/2 ) = (0,..., 0, 1,..., 1) is fixed by every wiring of every gate from Γ. Γ is universal on K = 2 r bits: Implementation of γ Γ 0 on bits 1, 2, 3: For T {4, 5,..., 2 r+1 + 2}, T = 2 r : γ T = γ, wired to 1, 2, 3 and T. In γ T, there are an odd number of T s allowing γ to act. T
8 Universality and invariants Universality and invariants Invariants and ideals Experimental results Orbits on 4-tuples Theorem (using CFSG): G S m G is 4 transitive G A m or G is Mathieu... Corollary (m {11, 12, 23, 24}) Γ is K-universal Γ + bit permutations has 15 orbits on 4-tuples dim of invariant vectors of Γ {bit permutations} in C 24K is 15 (Γ acts on the first n bits)
9 Invariants and ideals Universality and invariants Invariants and ideals Experimental results Bit slicing: B = C 2, W = B 4, invariants in W K Fixed points on the tensor algebra K=0 W K = Dual of a homogeneous ideal in the 2 4 -variable polynomial ring dual ideal because Γ acts on the first n levels commutative polynomial ring because of bit permutations The degree K level of the ideal is spanned by binomials corresponding the relation in the same orbit
10 Universality and invariants Invariants and ideals Experimental results dim of fixed points at level K Hilbert function Ultimately a polynomial, the Hilbert polynomial from which K?: regularity Hilbert polynomial=constant 15 zero dim. ideal D. Lazard s regularity bound gives approx. 16n.
11 Experimental results Universality and invariants Invariants and ideals Experimental results - GAP: 23 subgroups of S 8 containing the bit permutations are not 3-universal, but 4-universal. Smallest: order 72, not transitive. - C program + Macaulay 2: no further universal subgroups bits trits - action on triples sufficient (classification of 3-transitive groups) - the two-trit gates are universal - GAP (with Csaba Schneider) 1700 subgroups of S 9 containing the swap of trits are not 2-universal, but 3-universal. Smallest: order 6, abelian! - C program + Macaulay 2: no further universal subgroups
12 Quantum circuits Universality of reversible gates Quantum circuits A density criterion Jeandel s criterion aka Larsen s alternative bit (0 or 1) quantum bit (qubit): unit vector a 0 + b 1 in C 2 n-bit strings n-qubit systems: unit vectors in C 2n n-bit (reversible) gates n-qubit (quantum) gates: unitary transformations of C 2n K-universality: generating A 2 K approximating all the elements of PSU 2 K : = generating a dense subgroup The set of 2-qubit gates is universal. Jeandel s separation: originally quantum.
13 A density criterion Universality of reversible gates Quantum circuits A density criterion Jeandel s criterion aka Larsen s alternative Γ U 2 K generates (projectively) a dense subgroup of PSU 2 K The space of the vectors left invariant by the 2 8K 2 8K matrices γ γ γ γ γ γ γ γ (γ Γ ) has dimension 24. (Action on V 4 (V 4 ), V = C 2K ) (By classical invariant theory, 24 is the dimension of the invariants of the group (P)(S)U.) A consequence of a density criterion of Jeandel and a theorem of Guralnick and Tiep (2005) (latter uses CFSG). Universality bound: Lazard s regularity bound gives approx. 256n
14 Quantum circuits A density criterion Jeandel s criterion aka Larsen s alternative Jeandel s criterion aka Larsen s alternative Γ U 2 K generates a dense subgroup of PSU 2 K The following two conditions: (1) The space of the invariants of Γ on V 2 (V 2 ) has dimension 2, and (2) the group generated by Γ is infinite (test: Alla s talk) Proof: G = complex Zariski closure of the group gen. by Γ. (1) sl 2 K is irreducible under the conjugation action of G. Then either G 0 = {1} or G 0 = SL 2 K.
15 Better bounds? Universality of reversible gates Better bounds? Tensor resp. direct product structure Further questions Tight(er) bounds? Quantum: already for n = 2. Reversible: n = 4 hope to do soon. Special structure of the ideal: R: binomials corresponding to a partition R,Q: make use of tensor structure of W??? e.g., (Q): W = B 4 (B 4 ), W n inherits the decomposition W = Mat 16 (C), W n has a matrix multiplication as well. Fixed points: subalgebra. Is there a finite group ( U 2 n), containing the qubit permutations which is universal? Already for n = 2???
16 Tensor resp. direct product structure Better bounds? Tensor resp. direct product structure Further questions Galois connection between relations (Q) resp. subspaces of W n and groups R (Krasner): On Ω Ω 2 Ω 3..., S = S 1 S 2 S 3... is the system of invariant relations of a permutation group on Ω S is a Krasner-algebra, i.e, S is closed under union,intersection, complement (in Ω k ), permutation of coordinates, projection, extension, and contains the identity relation. Q (Schrijver): A subspace S = k,l = 0 Sk l of T (W ) T (W ) = k,l=0 W k W l is the system of tensor invariants of a group in GL(W ) iff it is a -subalgebra which is closed under tensor contractions and mutations (permutations of tensor-coordinates ), Q,R: Consequences to invariants on small powers? Q: Fixed points of finite groups???
17 Further questions Universality of reversible gates Better bounds? Tensor resp. direct product structure Further questions Can anything (e.g., decidability of universality) proved without CFSG???? R: Can one prove that for a constant m, infinitely many K s.t. on 2 K elements the m-transitive groups contain A 2 K? Ancillas (auxiliary memory) R: Elements of A 2 K+L mapping S = {(x 1,..., x K, 0,..., 0)} to itself (A 2 K 1) should be realized, action considered only on S. Q: Elements of U 2 K+L mapping S = C 2K to itself (U 2 K I ) should be realized, action considered only on S.
18 Approximate input (Q)? Better bounds? Tensor resp. direct product structure Further questions Closeness to a non-universal gate set is decidable. (Impractical algorithm, based on solving a system of polynomial inequalities.) Is there a more efficient method???? Is it true that if Γ on W n has approx. fixed point v of norm 1 then Γ is close to a set which has a fixed point close to v?
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