# 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, Sum, two
from proveit.physics.quantum.QPE import Pfail, _neg_domain, _pos_domain, _rel_indexed_alpha

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

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[P_{\rm fail}\right]\left(e\right) = \left(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\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: 5
operand: 48
4Operationoperator: 55
operands: 7
5Literal
6ExprTuple48
7ExprTuple8, 9
8Operationoperator: 11
operand: 13
9Operationoperator: 11
operand: 14
10ExprTuple13
11Literal
12ExprTuple14
13Lambdaparameter: 50
body: 15
14Lambdaparameter: 50
body: 17
15Conditionalvalue: 19
condition: 18
16ExprTuple50
17Conditionalvalue: 19
condition: 20
18Operationoperator: 23
operands: 21
19Operationoperator: 51
operands: 22
20Operationoperator: 23
operands: 24
21ExprTuple50, 25
22ExprTuple26, 53
23Literal
24ExprTuple50, 27
25Operationoperator: 31
operands: 28
26Operationoperator: 29
operand: 35
27Operationoperator: 31
operands: 32
28ExprTuple33, 34
29Literal
30ExprTuple35
31Literal
32ExprTuple41, 47
33Operationoperator: 55
operands: 36
34Operationoperator: 59
operand: 41
35Operationoperator: 38
operand: 42
36ExprTuple40, 61
37ExprTuple41
38Literal
39ExprTuple42
40Operationoperator: 59
operand: 47
41Operationoperator: 55
operands: 44
42Operationoperator: 45
operands: 46
43ExprTuple47
44ExprTuple48, 61
45Literal
46ExprTuple49, 50
47Operationoperator: 51
operands: 52
48Variable
49Literal
50Variable
51Literal
52ExprTuple53, 54
53Literal
54Operationoperator: 55
operands: 56
55Literal
56ExprTuple57, 58
57Literal
58Operationoperator: 59
operand: 61
59Literal
60ExprTuple61
61Literal