Geometric theory of nonlocal two-qubit operations

The geometric structure of non-local two-qubit gates is a 3-Torus.

“By applying a Cartan decomposition to su (4), we find that the geometric structure of non-local gates is a 3-Torus.”

• Also stated in: § A geometric theory of non-local two-qubit operations · p. 1 · plain

  “We find that the geometric structure of non-local gates is a 3-Torus.”

• Also stated in: § III. NON-LOCAL OPERATIONS · p. 6 · plain

  “It also reveals that the geometric structure of the local equivalence classes is none other than a 3-Torus.”

The Abelian subalgebra a is a Cartan subalgebra of the pair (g,k).

“Hence it is a Cartan subalgebra of the pair ( g , k ).”

The local invariants of U(t) for H2 are G1(t)=cos^4(t/2) and G2(t)=1+2 cos t.

“The local invariants of U ( t ) are G 1 ( t ) = cos 4 t 2 and G 2 ( t ) = 1 + 2 cos t .”

The isotropic Heisenberg exchange Hamiltonian H1 generates the flow [t/2,t/2,t/2] on the 3-Torus.

“Hence the Hamiltonian H 1 generates the flow [ t 2 , t 2 , t 2 ] on the 3-Torus.”

The corresponding point for ControlledU in a+ is [γ,0,0].

“we find that [ γ, 0 , 0] is the corresponding point in a + .”

The corresponding point for CNOT in the Weyl chamber a+ is [π/2,0,0].

“we find that [ π 2 , 0 , 0] is the corresponding point for CNOT in the Weyl chamber a + .”

For the ControlledU gate, the local invariants are G1=cos^2 γ and G2=2 cos^2 γ +1, where γ = sqrt(γ1^2+γ2^2+γ3^2).

“For the ControlledU gate, the local invariants are G 1 = cos 2 γ and G 2 = 2 cos 2 γ +1, where γ = √ γ 2 1 + γ 2 2 + γ 2 3 .”

The XY Hamiltonian H2 generates the flow [t/2,t/2,0] for t ∈ [2kπ, 2kπ+π], and [t/2, π - t/2, 0] for t ∈ [2kπ+π, 2(k+1)π].

“The Hamiltonian H 2 generates the flow [ t 2 , t 2 , 0] for t ∈ [2 kπ, 2 kπ + π ], and [ t 2 , π -t 2 , 0] for t ∈ [2 kπ + π, 2( k +1) π ].”

The corresponding point for √SWAP in a+ is [π/4,π/4,π/4].

“we derive that [ π 4 , π 4 , π 4 ] is the corresponding point in a + .”

The local invariants G1 and G2 are related to the coordinates [c1,c2,c3] of the 3-Torus structure of non-local two-qubit gates.

“Eq. (30) reveals the relation between the local invariants G 1 and G 2 and the coordinates [ c 1 , c 2 , c 3 ] of the 3-Torus structure of non-local two-qubit gates.”

A quantum circuit containing at most three non-local gates generated by a two-body interaction Hamiltonian and at most four local gates generated by single-qubit terms can simulate any arbitrary two-qubit gate exactly.

“We explicitly construct a quantum circuit that contains at most three non-local gates generated by a two-body interaction Hamiltonian, together with at most four local gates generated by single qubit terms.”

• Also stated in: § I. INTRODUCTION · p. 2 · boosted

  “We explicitly construct a quantum circuit that contains three non-local gates generated by a given two-body interaction Hamiltonian for corresponding finite time durations, together with at most four local gates. We prove that such a quantum circuit can simulate any arbitrary two-qubit operation exactly and is therefore universal.”

The local invariants of U(t) for H3 are G1(t)=cos^2(t/2) and G2(t)=2+cos t.

“The local invariants of U ( t ) are G 1 ( t ) = cos 2 t 2 and G 2 ( t ) = 2 + cos t .”

The trajectory generated by H3 in a+ is [t/2,0,0], which evolves along the line OA1.

“The trajectory generated by the Hamiltonian H 3 in a + is [ t 2 , 0 , 0], which evolves along the line OA 1 .”

In general, multiple points on the 3-Torus correspond to a given pair of local invariants G1 and G2.

“In general, we expect to find multiple points on the 3-Torus for a given pair G 1 and G 2 .”

For any arbitrary H = 1/2(c1 σxσx + c2 σyσy + c3 σzσz), the trajectory on the 3-Torus is [c1 t, c2 t, c3 t].

“For any arbitrary H = 1 2 ( c 1 σ 1 x σ 2 x + c 2 σ 1 y σ 2 y + c 3 σ 1 z σ 2 z ), the trajectory on the 3-Torus is [ c 1 t, c 2 t, c 3 t ].”

The local equivalence classes of two-qubit gates are in one-to-one correspondence with the points of a Weyl chamber.

“Since each orbit of the Weyl group W ( G,K ) on a contains precisely one point in a Weyl chamber, the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points of a Weyl chamber.”

Gates directly generated by a given Hamiltonian form a one-dimensional subset on the 3-Torus.

“Generally, these gates form a one dimensional subset on the 3-Torus.”

The decomposition g = k ⊕ p is a Cartan decomposition of su(4).

“Therefore the decomposition g = k ⊕ p is a Cartan decomposition of su (4).”

The corresponding point for SWAP is [π/2,π/2,π/2].

“we obtain that the corresponding point for SWAP is [ π 2 , π 2 , π 2 ]”

The local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base.

“We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base.”

• Also stated in: § III. NON-LOCAL OPERATIONS · p. 6 · plain

  “We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron, except on the base where there are two equivalent areas.”

• Also stated in: § B. Geometric representation of two-qubit gates · p. 9 · plain

  “Note that every point in a + corresponds to a different local equivalence class.”

The points in Corollary 2 correspond to the triangles LMN, LPQ, and NPA2, which are three faces of the set of perfect entanglers.

“The points in Corollary 2 correspond to the triangles LMN, LPQ, and NPA 2 , which are three faces of the set of perfect entanglers.”

If the Hamiltonian H contains the non-local part, then H can generate non-local gates.

“If the Hamiltonian H contains the non-local part, that is, iH ∩ p = ∅ , then H can generate non-local gates.”

If c2+c3 ≥ π/2, the gate is not a perfect entangler.

“If c 2 + c 3 ≥ π 2 , that is, all the e i 2( c j + c k ) are on the lower semi-circle, the gate is not a perfect entangler either.”

If c1+c2 ≤ π/2, the gate is not a perfect entangler.

“If c 1 + c 2 ≤ π 2 , that is, all the e i 2( c j + c k ) are on the upper semi-circle, the gate is not a perfect entangler.”

√SWAP and its inverse are the only two perfect entanglers that can be achieved by H1.

“Moreover, it can easily be seen that √ SWAP and its inverse are the only two perfect entanglers that can be achieved by this Hamiltonian.”

All ControlledU gates correspond to the line OL in a+, where L is [CNOT].

“Hence, all the ControlledU gates correspond to the line OL in a + , where L is [CNOT].”

Among all non-local two-qubit gates, half of them are perfect entanglers.

“This implies that among all the non-local two-qubit gates, half of them are perfect entanglers.”

A two-qubit gate U is a perfect entangler if and only if the convex hull of the eigenvalues of m(U) contains zero.

“Theorem 1 (Condition for perfect entangler) A two-qubit gate U is a perfect entangler if and only if the convex hull of the eigenvalues of m ( U ) contains zero.”

• cites [2] Y. Makhlin (2000), e-print quant-ph/0002045. sourceNot checked yet

  Not checked in the free run.

For a two-qubit gate U, if its corresponding point in a+ is [c1,π/2-c1,c3], [c1,c1-π/2,c3], or [c1,c2,π/2-c2], U is a perfect entangler.

“Corollary 2 For a two-qubit gate U , if its corresponding point in the Weyl chamber a + is [ c 1 , π 2 -c 1 , c 3 ] , [ c 1 , c 1 -π 2 , c 3 ] , or [ c 1 , c 2 , π 2 -c 2 ] , U is a perfect entangler.”

CNOT, located at L, is the only perfect entangler that can be generated by H3.

“As noted above, CNOT, located at L, is the only perfect entangler that can be generated by this Hamiltonian.”

The entangling property of a quantum gate is determined only by its geometric representation on the 3-Torus.

“We show that the entangling property of a quantum gate is only determined by its geometric representation on the 3-Torus.”

Whether a two-qubit gate is a perfect entangler is only determined by its geometric representation [c1,c2,c3] on the 3-Torus.

“The above analysis shows that whether a two-qubit gate is a perfect entangler or not is only determined by its geometric representation [ c 1 , c 2 , c 3 ] on the 3-Torus.”

Such a quantum circuit can efficiently provide maximal entanglement from any arbitrary Hamiltonian of this form.

“In particular, it can therefore efficiently provide maximal entanglement from any arbitrary Hamiltonian of this form.”

The global phase factor does not affect the entangling properties of a two-qubit gate.

“It is clear that the global phase factor just rotates the eigenvalues of m ( U ) along the unit circle in the complex plane, while keeping their relative phase invariant. Therefore, it does not affect the entangling properties and we can consequently divide by det( U ).”

Generally, the gates that can be generated by a Hamiltonian directly form a one dimensional subset on the 3-Torus geometric representation of non-local gates.

“Generally speaking, these gates form a one dimensional subset on the 3-Torus geometric representation of non-local gates.”

√SWAP is a perfect entangler.

“picking c 1 = π 4 and c 3 = π 4 , we get the perfect entangler [ √ SWAP].”

For a gate [c1,c2,c3] in a+, its inverse is [π/2 - c1, c2, c3].

“Note that for a gate [ c 1 , c 2 , c 3 ] in a + , its inverse is just [ π 2 -c 1 , c 2 , c 3 ].”

The coordinates [c1,c2,c3] in the Weyl chamber a+ provide an alternative set of local invariants that is entirely equivalent to G1 and G2.

“With this representation in the tetrahedron a + we have removed the multiple-valued nature of the coordinates on the 3-Torus and cube and therefore can now take the coordinates [ c 1 , c 2 , c 3 ] as an alternative set of local invariants. They provide a useful geometric representation of local invariants that is easy to visualize and is entirely equivalent to G 1 and G 2 .”

The trajectory generated by H2 evolves along OA2A1.

“Hence the trajectory evolves along OA 2 A 1 .”

Exactly half of all non-local two-qubit operations are perfect entanglers.

“We find that exactly half the non-local gates are perfect entanglers.”

• Also stated in: § I. INTRODUCTION · p. 2 · plain

  “This implies that amongst all the non-local two-qubit operations, exactly half of them are capable of generating maximal entanglement.”

The flow generated by H1 evolves along OA3A1, which corresponds to all the local equivalence classes that can be generated by H1.

“Therefore, the flow generated by the isotropic Hamiltonian H 1 evolves along OA 3 A 1 , which corresponds to all the local equivalence classes that can be generated by H 1 .”

H2 can generate a set of perfect entanglers that corresponds to the line segments QA2 and A2M in a+.

“It is evident that H 2 can generate a set of perfect entanglers that corresponds to the line segments QA 2 and A 2 M in a + .”

Gates represented by points in the set {X ∈ a+ | 2(c1+c3) ≥ 2(c2+c3)+π} are not perfect entanglers.

“From Theorem 2, we obtain that the gates represented by points in the set { X ∈ a + | 2( c 1 + c 3 ) ≥ 2( c 2 + c 3 ) + π } are not perfect entanglers.”

The gates generated by H3 are all the ControlledU gates.

“Hence the gates generated by the Hamiltonian H 3 are all the ControlledU gates.”

The polyhedron LMNPQA2 is the residual set of perfect entanglers.

“it is thereby evident that the polyhedron LMNPQA 2 is the residual set of perfect entanglers.”

For H = 1/2(c1 σxσx + c2 σyσy + c3 σzσz), the local equivalence classes of U(t) form a continuous flow on the 3-Torus as time evolves.

“Assume H = 1 2 ( c 1 σ 1 x σ 2 x + c 2 σ 1 y σ 2 y + c 3 σ 1 z σ 2 z ). The local equivalence classes of U ( t ) form a continuous flow on the 3-Torus as time evolves.”

CNOT, located at L, is the only ControlledU gate that is a perfect entangler.

“Hence CNOT, located at L, is the only ControlledU gate that is a perfect entangler.”

CNOT is a perfect entangler.

“picking c 1 = π 2 and c 3 = 0, we obtain the perfect entangler [CNOT]”

The set of perfect entanglers is obtained by removing the three tetrahedra LQPO, NPA2A3, and LMNA1 from OA1A2A3.

“The set of perfect entanglers can thus be obtained by removing these three tetrahedra from OA 1 A 2 A 3 .”

From the constructive proof, together with four appropriate local gates, we can simulate any arbitrary two-qubit gate by turning on a two-body interaction Hamiltonian for at most three times.

“From the above constructive proof, it is clear that together with four appropriate local gates, we can simulate any arbitrary two-qubit gate by turning on a two-body interaction Hamiltonian for at most three times.”

Given a two-body interaction Hamiltonian, it is always possible to explicitly construct a quantum circuit for exact simulation of any arbitrary non-local two-qubit gate by turning on the two-body interaction for at most three times, together with four local gates.

“We proved that given a two-body interaction Hamiltonian, it is always possible to explicitly construct a quantum circuit for exact simulation of any arbitrary non-local two-qubit gate by turning on the two-body interaction for at most three times, together with four local gates.”

The way to choose the local gates k0, k1, and k2 is not unique.

“Also note that in the proof, the way to choose the local gates k 0 , k 1 , and k 2 is not unique.”

By applying the Hamiltonian at most three times, together with four appropriate local gates, we can exactly simulate any arbitrary two-qubit gate, i.e., implement any SU(4) operation.

“We will show that by applying the Hamiltonian at most three times, together with four appropriate local gates, we can exactly simulate any arbitrary two-qubit gate, i.e. , we can implement any SU (4) operation.”

The trajectory of U(t) in the Weyl chamber a+ is [c1 t, c2 t, c3 t].

“Therefore, the trajectory of U ( t ) in the Weyl chamber a + is [ c 1 t, c 2 t, c 3 t ].”

The Hamiltonian H_J is not able to generate the perfect entangler [√SWAP].

“Therefore, the Hamiltonian H J is not able to generate the perfect entangler [ √ SWAP].”

Each point inside the tetrahedron corresponds to a different equivalent class of non-local gates.

“Each point inside this tetrahedron corresponds to a different equivalent class of non-local gates.”

By further reducing the symmetry, the geometric representation of non-local gates is a tetrahedron.

“By further reducing the symmetry, the geometric representation of non-local gates was seen to be conveniently visualized as a tetrahedron.”

Since c3=0, the Hamiltonian H_J can reach only those local equivalence classes on the base OA1A2.

“Since c 3 = 0, the Hamiltonian H J can reach only those local equivalence classes on the base OA 1 A 2 , as shown in Figure 4.”

H and H_a have the same set of eigenvalues.

“Eq. (58) also implies that H and H a have the same set of eigenvalues.”

[CNOT] can be achieved by a circuit consisting of two √SWAP and a local gate.

“[CNOT] can however be achieved by a circuit consisting of two √ SWAP and a local gate [32].”

• cites [32] G. Burkard, D. Loss, D. P. DiVincenzo, and J. A. Smolin, Phys. Rev. B 60 , 11404 (1999). sourceNot checked yet

  Not checked in the free run.

When the Hamiltonian H contains both the local and non-local part, the velocity of the flow on the 3-Torus is usually time dependent.

“However, when the Hamiltonian H contains both the local and non-local part, the velocity of the flow on the 3-Torus is usually time dependent as shown in the above example.”

The geometric structure of non-local gates is a 3-Torus.

“We first showed that the geometric structure of non-local gates is a 3-Torus.”

Given a Hamiltonian H with iH ∈ p, any arbitrary two-qubit gate U ∈ SU(4) can be simulated by the quantum circuit (79) where k_j are local gates.

“Theorem 3 (Universal quantum circuit) Given a Hamiltonian H with iH ∈ p , any arbitrary two-qubit gate U ∈ SU (4) can be simulated by the following quantum circuit”

This guarantees that a highly efficient simulation of non-local gates can be made with any Hamiltonian consisting of arbitrary two-qubit interactions and allowing control of single qubit operations.

“This guarantees that a highly efficient simulation of non-local gates can be made with any Hamiltonian consisting of arbitrary two-qubit interactions and allowing control of single qubit operations.”