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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
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 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 = InverseFourierTransform(_t)
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = 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_expr2, sub_expr1, sub_expr3), ScalarMult(sub_expr2, Qmult(sub_expr1, sub_expr3))).with_wrapping_at(2)
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())
\begin{array}{c} \begin{array}{l} \left(\frac{1}{2^{\frac{t}{2}}} \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) =  \\ \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) \end{array} \end{array}
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.()(2)('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: 10
operands: 5
4Operationoperator: 28
operands: 6
5ExprTuple7, 13, 14
6ExprTuple7, 8
7Operationoperator: 22
operands: 9
8Operationoperator: 10
operands: 11
9ExprTuple60, 12
10Literal
11ExprTuple13, 14
12Operationoperator: 54
operands: 15
13Operationoperator: 16
operand: 59
14Operationoperator: 18
operand: 21
15ExprTuple58, 20
16Literal
17ExprTuple59
18Literal
19ExprTuple21
20Operationoperator: 22
operands: 23
21Lambdaparameter: 51
body: 25
22Literal
23ExprTuple59, 58
24ExprTuple51
25Conditionalvalue: 26
condition: 27
26Operationoperator: 28
operands: 29
27Operationoperator: 30
operands: 31
28Literal
29ExprTuple32, 33
30Literal
31ExprTuple51, 34
32Operationoperator: 54
operands: 35
33Operationoperator: 36
operands: 37
34Operationoperator: 38
operands: 39
35ExprTuple40, 41
36Literal
37ExprTuple51, 59
38Literal
39ExprTuple42, 43
40Literal
41Operationoperator: 44
operands: 45
42Literal
43Operationoperator: 46
operands: 47
44Literal
45ExprTuple58, 48, 49, 50, 51
46Literal
47ExprTuple52, 53
48Literal
49Literal
50Literal
51Variable
52Operationoperator: 54
operands: 55
53Operationoperator: 56
operand: 60
54Literal
55ExprTuple58, 59
56Literal
57ExprTuple60
58Literal
59Literal
60Literal