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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.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 = [k]
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))
expr = Equals(Mult(sub_expr2, Sum(index_or_indices = sub_expr1, summand = sub_expr3, domain = _m_domain)), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr3), domain = _m_domain)).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(\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) =  \\ \left(\sum_{k = 0}^{2^{t} - 1} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \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: 54
operands: 5
4Operationoperator: 9
operand: 8
5ExprTuple18, 7
6ExprTuple8
7Operationoperator: 9
operand: 12
8Lambdaparameter: 60
body: 11
9Literal
10ExprTuple12
11Conditionalvalue: 13
condition: 17
12Lambdaparameter: 60
body: 15
13Operationoperator: 54
operands: 16
14ExprTuple60
15Conditionalvalue: 19
condition: 17
16ExprTuple18, 19
17Operationoperator: 20
operands: 21
18Operationoperator: 49
operands: 22
19Operationoperator: 54
operands: 23
20Literal
21ExprTuple60, 24
22ExprTuple51, 25
23ExprTuple26, 27
24Operationoperator: 28
operands: 29
25Operationoperator: 56
operands: 30
26Operationoperator: 56
operands: 31
27Operationoperator: 56
operands: 32
28Literal
29ExprTuple33, 34
30ExprTuple62, 35
31ExprTuple37, 36
32ExprTuple37, 38
33Literal
34Operationoperator: 39
operands: 40
35Operationoperator: 49
operands: 41
36Operationoperator: 54
operands: 42
37Literal
38Operationoperator: 47
operand: 46
39Literal
40ExprTuple53, 44
41ExprTuple63, 62
42ExprTuple62, 58, 59, 45, 60
43ExprTuple46
44Operationoperator: 47
operand: 51
45Literal
46Operationoperator: 49
operands: 50
47Literal
48ExprTuple51
49Literal
50ExprTuple52, 53
51Literal
52Operationoperator: 54
operands: 55
53Operationoperator: 56
operands: 57
54Literal
55ExprTuple62, 58, 59, 60, 61
56Literal
57ExprTuple62, 63
58Literal
59Literal
60Variable
61Variable
62Literal
63Literal