# 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.
# import Expression classes needed to build the expression
from proveit import e, l
from proveit.logic import Equals
from proveit.numbers import Abs, Add, Exp, Interval, Neg, Sum, one, subtract, two
from proveit.physics.quantum.QPE import _neg_domain, _rel_indexed_alpha, _two_pow__t_minus_one

In [2]:
# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
expr = Equals(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _neg_domain), Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(Add(Neg(_two_pow__t_minus_one), one), subtract(Neg(e), one))))

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(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) = \left(\sum_{l = -2^{t - 1} + 1}^{-e - 1} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\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: 6
operand: 8
4Operationoperator: 6
operand: 9
5ExprTuple8
6Literal
7ExprTuple9
8Lambdaparameter: 47
body: 10
9Lambdaparameter: 47
body: 12
10Conditionalvalue: 14
condition: 13
11ExprTuple47
12Conditionalvalue: 14
condition: 15
13Operationoperator: 18
operands: 16
14Operationoperator: 50
operands: 17
15Operationoperator: 18
operands: 19
16ExprTuple47, 20
17ExprTuple21, 52
18Literal
19ExprTuple47, 22
20Operationoperator: 26
operands: 23
21Operationoperator: 24
operand: 29
22Operationoperator: 26
operands: 27
23ExprTuple30, 28
24Literal
25ExprTuple29
26Literal
27ExprTuple30, 31
28Operationoperator: 58
operand: 37
29Operationoperator: 33
operand: 38
30Operationoperator: 54
operands: 35
31Operationoperator: 54
operands: 36
32ExprTuple37
33Literal
34ExprTuple38
35ExprTuple39, 60
36ExprTuple40, 57
37Operationoperator: 54
operands: 41
38Operationoperator: 42
operands: 43
39Operationoperator: 58
operand: 48
40Operationoperator: 58
operand: 49
41ExprTuple49, 60
42Literal
43ExprTuple46, 47
44ExprTuple48
45ExprTuple49
46Literal
47Variable
48Operationoperator: 50
operands: 51
49Variable
50Literal
51ExprTuple52, 53
52Literal
53Operationoperator: 54
operands: 55
54Literal
55ExprTuple56, 57
56Literal
57Operationoperator: 58
operand: 60
58Literal
59ExprTuple60
60Literal