Fermionic simulation gates
Methods, systems, and apparatus for simulating a physical system. A Hamiltonian describing the physical system is transformed into a qubit Hamiltonian describing a corresponding system of qubits, the qubit Hamiltonian comprising a transformed kinetic energy operator. The evolution of the system of qubits under the qubit Hamiltonian is simulated, including simulating the evolution of the system of qubits under the transformed kinetic energy operator by applying a fermionic swap network to the system of qubits. The simulated evolution of the system of qubits under the qubit Hamiltonian is used to determine properties of the physical system.
1 . A method performed by a quantum computer comprising a system of qubits with linear nearest neighbor connectivity, the method comprising:
implementing a Trotter step of evolution of the system of qubits under a qubit Hamiltonian by applying a fermionic swap network to the system of qubits, wherein applying the fermionic swap network comprises:
applying a plurality of layers of fermionic simulation gates to the system of qubits to simulate a kinetic energy operator included in the qubit Hamiltonian, wherein each fermionic simulation gate is a quantum logic gate that operates on two qubits and is configured to perform, within a single gate operation, (i) a fermionic mode exchange between the two qubits and (ii) time evolution of the kinetic energy operator acting between the two qubits,
applying multiple single qubit gates to the system of qubits to simulate single qubit terms in an external potential term and single qubit terms arising from an interaction term included in the qubit Hamiltonian, wherein the single qubit gates are interleaved between successive layers of the fermionic simulation gates; and
measuring the system of qubits to determine an output of the Trotter step of evolution.
2 . The method of claim 1 , further comprising implementing multiple Trotter steps of evolution to perform a variational algorithm and obtain a quantum state that is a variational approximation to a target quantum state.
3 . The method of claim 1 , wherein the fermionic swap network comprises a quantum circuit comprising multiple layers of fermionic swap gates.
4 . The method of claim 3 , wherein the qubits in the system of qubits are indexed according to a canonical ordering, and wherein implementing the Trotter step of evolution of the system of qubits comprises sequentially applying each of the multiple layers of fermionic swap gates to the system of qubits to change the canonical ordering of the qubits until the canonical ordering is reversed and each index has been adjacent to all others once.
5 . The method of claim 4 , wherein the kinetic energy operator comprises one or more operators that act on multiple non-adjacent qubits and wherein sequentially applying each of the multiple layers of fermionic swap gates to the system of qubits maps the one or more operators that act on multiple non-adjacent qubits to operators that act on two neighboring qubits.
6 . The method of claim 5 , wherein the one or more operators that act on multiple non-adjacent qubits comprise operators of the form X p Z p+1 Z p+2 . . . Z q−1 X q and Y p Z p+1 Z p+2 . . . Z q−1 Y q , with X p representing a Pauli-X operator applied to qubit p, Y p representing a Pauli-Y operator applied to qubit p, and Z p representing a Pauli-Z operator applied to qubit p, and wherein the operators that act on two neighboring qubits comprise operators of the form X p X p+1 and Y p Y p+1 .
7 . The method of claim 6 , wherein applying the fermionic swap network further comprises:
interleaving gates for simulating the evolution of the system of qubits under the operators that act on two neighboring qubits between layers of the fermionic swap network; and
applying the interleaved gates when applying the fermionic swap network to the system of qubits.
8 . The method of claim 6 , wherein sequentially applying each of the multiple layers of fermionic swap gates to the system of qubits to change the canonical ordering of the qubits comprises:
indexing the qubits by the canonical ordering from 1 to N;
sequentially applying:
fermionic swap gates between odd numbered qubits and even numbered qubits to the right; and
fermionic swap gates between even numbered qubits and odd numbered qubits to the right.
9 . The method of claim 8 , wherein the qubit Hamiltonian comprises a Jordan-Wigner transform of a corresponding Hamiltonian that describes a physical system, the physical system comprising a system of electrons and wherein the total number of applied layers comprises N applied layers, with N representing the number of orbitals in the system of electrons and the number of qubits in the system of qubits.
10 . The method of claim 1 , wherein the qubit Hamiltonian comprises a Jordan-Wigner transform of a corresponding Hamiltonian that describes a physical system.
11 . The method of claim 10 , wherein the physical system comprises a system of electrons or is described by a two-dimensional Hubbard model.
12 . The method of claim 1 , wherein the interaction term comprises operators of the form ( −Z p −Z p+1 +Z p Z p+1 ) where Z p represents a Pauli-Z operator applied to qubit p.
13 . The method of claim 1 , wherein each fermionic simulation gate is configured to approximately simultaneously
(i) simulate the evolution of the system of qubits under the operator X p X p+1 +Y p Y p+1 for a time φ, where X p represents a Pauli-X operator applied to qubit p and Y p represents a Pauli-Y operator applied to qubit p
(ii) simulate evolution of the system of qubits under the operator Z p Z P+1 for a time θ, where Z p represents a Pauli-Z operator applied to qubit p, and
(iii) apply a fermionic swap gate to qubits p and p+1.
14 . The method of claim 13 , wherein the fermionic simulation gate is given by
ℱ
(
θ
,
ϕ
)
=
exp
(
-
i
[
π
4
(
Z
p
+
Z
p
+
1
)
+
(
ϕ
+
π
4
)
(
X
p
X
p
+
1
+
Y
p
Y
p
+
1
)
+
θ
Z
p
Z
p
+
1
-
π
2
]
)
.
15 . The method of claim 1 , wherein N choose 2 fermionic simulation gates are sufficient to implement the Trotter step of evolution, where N represents the number of qubits in the system of qubits.
16 . The method of claim 1 , wherein a fermionic swap gate acting on qubit p and qubit q=p+1 is given by
f
swap
=
Jordan
Wigner
[
f
swap
p
,
q
]
=
(
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
-
1
)
.
17 . An apparatus comprising:
quantum hardware, comprising:
a system of qubits with linear nearest neighbor connectivity,
a plurality of single qubit gates,
a plurality of two qubit gates;
one or more classical processors;
wherein the apparatus is configured to perform operations comprising:
implementing a Trotter step of evolution of the system of qubits under a qubit Hamiltonian by applying a fermionic swap network to the system of qubits, wherein applying the fermionic swap network comprises:
applying a plurality of layers of fermionic simulation gates to the system of qubits to simulate a kinetic energy operator included in the qubit Hamiltonian, wherein each fermionic simulation gate is a quantum logic gate that operates on two qubits and is configured to perform, within a single gate operation, (i) a fermionic mode exchange between the two qubits and (ii) time evolution of the kinetic energy operator acting between the two qubits,
applying multiple single qubit gates to the system of qubits to simulate single qubit terms in an external potential term and single qubit terms arising from an interaction term included in the qubit Hamiltonian, wherein the single qubit gates are interleaved between successive layers of the fermionic simulation gates; and
measuring the system of qubits to determine an output of the Trotter step of evolution.