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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

In [1]:
import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import k, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, ScalarMult(sub_expr3, Qmult(sub_expr2, sub_expr4))), Qmult(sub_expr3, sub_expr1, sub_expr2, sub_expr4))
expr:
In [3]:
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")
Passed sanity check: expr matches stored_expr
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\left({_{t}}\langle m \rvert \thinspace \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right)\right) = \left(\frac{1}{2^{\frac{t}{2}}} \thinspace {_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()()('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 16
operands: 5
4Operationoperator: 16
operands: 6
5ExprTuple8, 7
6ExprTuple12, 8, 19, 20
7Operationoperator: 34
operands: 9
8Operationoperator: 10
operands: 11
9ExprTuple12, 13
10Literal
11ExprTuple14, 65
12Operationoperator: 28
operands: 15
13Operationoperator: 16
operands: 17
14Variable
15ExprTuple66, 18
16Literal
17ExprTuple19, 20
18Operationoperator: 60
operands: 21
19Operationoperator: 22
operand: 65
20Operationoperator: 24
operand: 27
21ExprTuple64, 26
22Literal
23ExprTuple65
24Literal
25ExprTuple27
26Operationoperator: 28
operands: 29
27Lambdaparameter: 57
body: 31
28Literal
29ExprTuple65, 64
30ExprTuple57
31Conditionalvalue: 32
condition: 33
32Operationoperator: 34
operands: 35
33Operationoperator: 36
operands: 37
34Literal
35ExprTuple38, 39
36Literal
37ExprTuple57, 40
38Operationoperator: 60
operands: 41
39Operationoperator: 42
operands: 43
40Operationoperator: 44
operands: 45
41ExprTuple46, 47
42Literal
43ExprTuple57, 65
44Literal
45ExprTuple48, 49
46Literal
47Operationoperator: 50
operands: 51
48Literal
49Operationoperator: 52
operands: 53
50Literal
51ExprTuple64, 54, 55, 56, 57
52Literal
53ExprTuple58, 59
54Literal
55Literal
56Literal
57Variable
58Operationoperator: 60
operands: 61
59Operationoperator: 62
operand: 66
60Literal
61ExprTuple64, 65
62Literal
63ExprTuple66
64Literal
65Literal
66Literal