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BraAndKet is a library for numeral calculations of discrete quantum systems.

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BraAndKet

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BraAndKet is a library for numeral calculations of discrete quantum systems.

Quickstart

Before Using

Please notice that this library is still actively developing. The stability and compatibility of APIs are NOT guaranteed. Breaking changes are happening every day! Using this library right now, you may take your own risk.

Installation

You can install the latest release from PiPy.

pip install braandket

Then you can import this library with name bnk

import braandket as bnk

KetSpace

Any quantum states can exist in some space called Hilbert space. You can use bnk.KetSpace(n) to define such a space, where n is its dimension. For example, to create a Hilbert space of a q-bit:

qbit = bnk.KetSpace(2)
print(qbit)  # output: KetSpace(2)

You can define a name for a space using named parameter. The name is to describe this space when debugging. The name can be a str, or any object to be printed out. When printed, the name of space will be shown, which is very helpful when debugging.

qbit_a = bnk.KetSpace(2, name="a")
print(qbit_a)  # output: KetSpace(2, name=a)

qbit_b = bnk.KetSpace(2, name="b")
print(qbit_b)  # output: KetSpace(2, name=b)

You can call these 4 methods on a KetSpace instance to create ket vectors and operators:

  • method .eigenstate(k) - to get a ket vector, representing the k-th eigenstate
  • method .identity() - to get an identity operator in this Hilbert space
  • method .operator(k,b) - to get an operator
  • method .projector(k) - to get a projector
ket_space = bnk.KetSpace(2)

ket_vec = ket_space.eigenstate(0)
identity_op = ket_space.identity()
increase_op = ket_space.operator(1, 0)
zero_proj = ket_space.projector(0)

A KetSpace is accompanied by a BraSpace. You can conveniently get it with .ct property. To avoid confusion, is not allowed to create any vectors or operations with a BraSpace. Please do so with its corresponding KetSpace. Calling .ct property, you can get back its KetSpace.

ket_space = bnk.KetSpace(2)
print(ket_space)  # output: KetSpace(2)

bra_space = ket_space.ct
print(bra_space)  # output: BraSpace(2)

print(bra_space.ct is ket_space)  # output: True

QTensors

QTensor is the basic type of computing elements in this library. A QTensor instance holds an np.ndarray as its values and a tuple of Space instances. Each Space corresponds to an axis of the np.ndarray.

Any vectors, operators and tensors in quantum world are represented by QTensor. All vectors and operators mentioned above are all QTensor instances.

ket_space = bnk.KetSpace(2)

ket_vec = ket_space.eigenstate(0)
print(ket_vec)
# output: QTensor(spaces=(KetSpace(2),), values=[1. 0.])

identity_op = ket_space.identity()
print(identity_op)
# output: QTensor(spaces=(KetSpace(2), BraSpace(2)), values=[[1. 0.] [0. 1.]])

increase_op = ket_space.operator(1, 0)
print(increase_op)
# output: QTensor(spaces=(KetSpace(2), BraSpace(2)), values=[[0. 0.] [1. 0.]])

zero_proj = ket_space.projector(0)
print(zero_proj)
# output: QTensor(spaces=(KetSpace(2), BraSpace(2)), values=[[1. 0.] [0. 0.]])

You can easily get a conjugate transposed QTensor calling .ct property. It should be noted that sometimes, such operation does not affect the values, but spaces.

ket_space = bnk.KetSpace(2)

ket_vec = ket_space.eigenstate(0)
bra_vec = ket_vec.ct
print(bra_vec)
# output: QTensor(spaces=(BraSpace(2),), values=[1. 0.])

increase_op = ket_space.operator(1, 0)
decrease_op = increase_op.ct
print(decrease_op)
# output: QTensor(spaces=(BraSpace(2), KetSpace(2)), values=[[0. 0.] [1. 0.]])

QTensor instances can take tensor product using @ operator. They can automatically inspect which spaces to be performed the "product-sum" (when the bra on the left meets the matching ket on the right), which to be remained.

Example1:

qbit = bnk.KetSpace(2)

amp = qbit.eigenstate(0).ct @ qbit.eigenstate(1)
print(amp)
# output: QTensor(spaces=(), values=0.0)

Example2:

qbit_a = bnk.KetSpace(2, name="a")
qbit_b = bnk.KetSpace(2, name="b")

ket_vec_ab = qbit_a.eigenstate(0) @ qbit_b.eigenstate(1)
print(ket_vec_ab)
# output: QTensor(spaces=(KetSpace(2, name=a), KetSpace(2, name=b)), values=[[0. 1.] [0. 0.]])

Example3:

qbit_a = bnk.KetSpace(2, name="a")
qbit_b = bnk.KetSpace(2, name="b")

tensor_ab = qbit_a.eigenstate(0).ct @ qbit_b.eigenstate(1)
print(tensor_ab)
# output: QTensor(spaces=(BraSpace(2, name=a), KetSpace(2, name=b)), values=[[0. 1.] [0. 0.]])

Example4:

qbit = bnk.KetSpace(2)

ket_vec_0 = qbit.eigenstate(0)
ket_vec_1 = qbit.eigenstate(1)
increase_op = qbit.operator(1, 0)
result = increase_op @ ket_vec_0

print(result)
# output: QTensor(spaces=(KetSpace(2),), values=[0. 1.])

print(result == ket_vec_1)
# output: True

(todo ...)

Pruning

Sometimes, the space of system can be terribly big, since the space increases exponentially with the increase of the count of components.

But in some cases, we just want to study the evolution of the system under certain conditions, for example from several specified start points evolves with some certain operators. Then, some states are in fact impossible to be reached. Then those unreachable states can be dropped out of the computation. Class PrunedKetSpace is designed for such cases.

The static method PrunedKetSpace.from_seed() can automatically detect which eigenstates can be dropped, with the given starting states and evolution operators, and return an instance of PrunedKetSpace as a "reachable" space. This can significantly reduce the calculation and memory consumption.

The pruned and original tensors can also be easily converted to each other using method reduce() and inflate().

Evolve functions

(todo ...)

Contribution

This library is completely open source. Any contributions are welcomed. You can fork this repository, make some useful changes and then send a pull request to me on GitHub.

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