logo

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.logic import Equals
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = frac(one, Exp(two, frac(_t, two)))
sub_expr2 = Sum(index_or_indices = [k], summand = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))), domain = _m_domain)
expr = Equals(Mult(sub_expr1, Mult(sub_expr1, sub_expr2)), Mult(sub_expr1, sub_expr1, sub_expr2)).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}}} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\right)\right) =  \\ \left(\frac{1}{2^{\frac{t}{2}}} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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: 52
operands: 5
4Operationoperator: 52
operands: 6
5ExprTuple9, 7
6ExprTuple9, 9, 10
7Operationoperator: 52
operands: 8
8ExprTuple9, 10
9Operationoperator: 45
operands: 11
10Operationoperator: 12
operand: 15
11ExprTuple51, 14
12Literal
13ExprTuple15
14Operationoperator: 54
operands: 16
15Lambdaparameter: 58
body: 18
16ExprTuple60, 19
17ExprTuple58
18Conditionalvalue: 20
condition: 21
19Operationoperator: 45
operands: 22
20Operationoperator: 52
operands: 23
21Operationoperator: 24
operands: 25
22ExprTuple61, 60
23ExprTuple26, 27
24Literal
25ExprTuple58, 28
26Operationoperator: 54
operands: 29
27Operationoperator: 54
operands: 30
28Operationoperator: 31
operands: 32
29ExprTuple34, 33
30ExprTuple34, 35
31Literal
32ExprTuple36, 37
33Operationoperator: 52
operands: 38
34Literal
35Operationoperator: 47
operand: 43
36Literal
37Operationoperator: 40
operands: 41
38ExprTuple60, 56, 57, 42, 58
39ExprTuple43
40Literal
41ExprTuple50, 44
42Literal
43Operationoperator: 45
operands: 46
44Operationoperator: 47
operand: 51
45Literal
46ExprTuple49, 50
47Literal
48ExprTuple51
49Operationoperator: 52
operands: 53
50Operationoperator: 54
operands: 55
51Literal
52Literal
53ExprTuple60, 56, 57, 58, 59
54Literal
55ExprTuple60, 61
56Literal
57Literal
58Variable
59Variable
60Literal
61Literal