# 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 Conditional, Lambda, e, l
from proveit.logic import Equals, InSet
from proveit.numbers import Abs, Add, Exp, Interval, Neg, 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 = Exp(Abs(_rel_indexed_alpha), two)
expr = Equals(Lambda(l, Conditional(sub_expr1, InSet(l, _neg_domain))), Lambda(l, Conditional(sub_expr1, InSet(l, Interval(Add(Neg(_two_pow__t_minus_one), one), subtract(Neg(e), one)))))).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[l \mapsto \left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\}\right..\right] =  \\ \left[l \mapsto \left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-e - 1\}\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
3Lambdaparameter: 42
body: 5
4Lambdaparameter: 42
body: 7
5Conditionalvalue: 9
condition: 8
6ExprTuple42
7Conditionalvalue: 9
condition: 10
8Operationoperator: 13
operands: 11
9Operationoperator: 45
operands: 12
10Operationoperator: 13
operands: 14
11ExprTuple42, 15
12ExprTuple16, 47
13Literal
14ExprTuple42, 17
15Operationoperator: 21
operands: 18
16Operationoperator: 19
operand: 24
17Operationoperator: 21
operands: 22
18ExprTuple25, 23
19Literal
20ExprTuple24
21Literal
22ExprTuple25, 26
23Operationoperator: 53
operand: 32
24Operationoperator: 28
operand: 33
25Operationoperator: 49
operands: 30
26Operationoperator: 49
operands: 31
27ExprTuple32
28Literal
29ExprTuple33
30ExprTuple34, 55
31ExprTuple35, 52
32Operationoperator: 49
operands: 36
33Operationoperator: 37
operands: 38
34Operationoperator: 53
operand: 43
35Operationoperator: 53
operand: 44
36ExprTuple44, 55
37Literal
38ExprTuple41, 42
39ExprTuple43
40ExprTuple44
41Literal
42Variable
43Operationoperator: 45
operands: 46
44Variable
45Literal
46ExprTuple47, 48
47Literal
48Operationoperator: 49
operands: 50
49Literal
50ExprTuple51, 52
51Literal
52Operationoperator: 53
operand: 55
53Literal
54ExprTuple55
55Literal