# Expressions in Q#

## Numeric Expressions

Numeric expressions are expressions of type Int, BigInt, or Double. That is, they are either integer or floating-point numbers.

Int literals in Q# are written as a sequence of digits. Hexadecimal and binary integers are supported and written with a 0x and 0b prefix, respectively.

BigInt literals in Q# have a trailing l or L suffix. Hexadecimal big integers are supported and written with a "0x" prefix. Thus, the following are all valid uses of BigInt literals:

let bigZero = 0L;
let bigHex = 0x123456789abcdef123456789abcdefL;
let bigOne = bigZero + 1L;


Double literals in Q# are floating-point numbers written using decimal digits. They can be written with or without a decimal point, ., or an exponential part indicated with 'e' or 'E' (after which only a possible negative sign and decimal digits are valid). The following are valid Double literals: 0.0, 1.2e5, 1e-5.

Given an array expression of any element type, you can form an Int expression using the Length built-in function, with the array expression enclosed in parentheses. For example, if a is bound to an array, then Length(a) is an integer expression. If b is an array of arrays of integers, Int[][], then Length(b) is the number of sub-arrays in b, and Length(b) is the number of integers in the second sub-array in b.

Given two numeric expressions of the same type, the binary operators +, -, *, and / may be used to form a new numeric expression. The type of the new expression is the same as the types of the constituent expressions.

Given two integer expressions, use the binary operator ^ (power) to form a new integer expression. Similarly, you can also use ^ with two double expressions to form a new double expression. Finally, you can use ^ with a big integer on the left and an integer on the right to form a new big integer expression. In this case, the second parameter must fit into 32 bits; if not, it raises a runtime error.

Given two integer or big integer expressions, form a new integer or big integer expression using the % (modulus), &&& (bitwise AND), ||| (bitwise OR), or ^^^ (bitwise XOR) operators.

Given either an integer or big integer expression on the left, and an integer expression on the right, use the <<< (arithmetic left shift) or >>> (arithmetic right shift) operators to create a new expression with the same type as the left-hand expression.

The second parameter (the shift amount) to either shift operation must be greater than or equal to zero; the behavior for negative shift amounts is undefined. The shift amount for either shift operation must also fit into 32 bits; if not, it raises a runtime error. If the number shifted is an integer, then the shift amount is interpreted mod 64; that is, a shift of 1 and a shift of 65 have the same effect.

For both integer and big integer values, shifts are arithmetic. Shifting a negative value either left or right results in a negative number. That is, shifting one step to the left or right is the same as multiplying or dividing by 2, respectively.

Integer division and integer modulus follow the same behavior for negative numbers as C#. That is, a % b always has the same sign as a, and b * (a / b) + a % b always equals a. For example:

A B A / B A % B
5 2 2 1
5 -2 -2 1
-5 2 -2 -1
-5 -2 2 -1

Big integer division and modulus operations work the same way.

Given any numeric expression, you can form a new expression using the - unary operator. The new expression is the same type as the constituent expression.

Given any integer or big integer expression, you can form a new expression of the same type using the ~~~ (bitwise complement) unary operator.

## Boolean Expressions

The two Bool literal values are true and false.

Given any two expressions of the same primitive type, the == and != binary operators may be used to construct a Bool expression. The expression is true if the two expressions are equal and false if not.

Values of user-defined types may not be compared, only their unwrapped values can be compared. For example, using the "unwrap" operator ! (explained in detail at Types in Q#),

newtype WrappedInt = Int;     // Yes, this is a contrived example
let x = WrappedInt(1);
let y = WrappedInt(2);
let z = x! == y!;             // This will compile and yield z = false.
let t = x == y;               // This will cause a compiler error.


Equality comparison for Qubit values is identity equality; that is, whether the two expressions identify the same qubit. The states of the two qubits are not compared, accessed, measured, or modified by this comparison.

Equality comparison for Double values may be misleading due to rounding effects. For example, 49.0 * (1.0/49.0) != 1.0.

Given any two numeric expressions, the binary operators >, <, >=, and <= may be used to construct a new Boolean expression that is true if the first expression is respectively greater than, less than, greater than or equal to, or less than or equal to the second expression.

Given any two Boolean expressions, use the and binary operator to construct a new Boolean expression that is true if both of the two expressions are true. Likewise, using the or operator creates an expression that is true if either of the two expressions is true.

Given any Boolean expression, the not unary operator may be used to construct a new Boolean expression that is true if the constituent expression is false.

## String expressions

Q# allows strings to be used in the fail statement (explained in Control Flow) and in the Message standard function. The specific behavior of the latter depends on the simulator used but typically writes a message to the host console when called during a Q# program.

Strings in Q# are either literals or interpolated strings.

String literals are similar to simple string literals in most languages: a sequence of Unicode characters enclosed in double-quotes " ". Inside of a string, use the backslash character \ to escape a double-quote character (\"), or to insert a new-line ( \n ), a carriage return (\r), or a tab (\t). For example:

"\"Hello world!\", she said.\n"


### Interpolated strings

The Q# syntax for string interpolations is a subset of the C# syntax. Following are the key points as they pertain to Q#:

• To identify a string literal as an interpolated string, prepend it with the $ symbol. There can be no white space between the $ and the " that starts a string literal.

• The following is a basic example using the Message function to write the result of a measurement to the console, alongside other Q# expressions.

    let num = 8;       // some Q# expression
let res = M(q);
Message(\$"Number: {num}, Result: {res}");

• Any valid Q# expression may appear in an interpolated string.

• Expressions inside of an interpolated string follow Q# syntax, not C# syntax. The most notable distinction is that Q# does not support verbatim (multi-line) interpolated strings.

For more details about the C# syntax, see Interpolated Strings.

## Range Expressions

Given any three Int expressions start, step, and stop, the expression start .. step .. stop is a range expression whose first element is start, second element is start+step, third element is start+step+step, and so on, until you pass stop. A range may be empty if, for example, step is positive and stop < start.

The range is inclusive at both ends. That is, if the difference between start and stop is an integer multiple of step, the last element of the range will be stop.

Given any two Int expressions start and stop, the expression start .. stop is a range expression that is equal to start .. 1 .. stop. Note that the implied step is +1 even if stop is less than start; in such a case, the range is empty.

Some example ranges are:

• 1..3 is the range 1, 2, 3.
• 2..2..5 is the range 2, 4.
• 2..2..6 is the range 2, 4, 6.
• 6..-2..2 is the range 6, 4, 2.
• 2..1 is the empty range.
• 2..6..7 is the range 2.
• 2..2..1 is the empty range.
• 1..-1..2 is the empty range.

## Qubit Expressions

The only Qubit expressions are symbols that are bound to Qubit values or array elements of Qubit arrays. There are no Qubit literals.

## Pauli Expressions

The four Pauli values, PauliI, PauliX, PauliY, and PauliZ, are all valid Pauli expressions.

Other than that, the only Pauli expressions are symbols that are bound to Pauli values or array elements of Pauli arrays.

## Result Expressions

The two Result values, One and Zero, are valid Result expressions.

Other than that, the only Result expressions are symbols that are bound to Result values or array elements of Result arrays. In particular, note that One is not the same as the integer 1, and there is no direct conversion between them. The same is true for Zero and 0.

## Tuple Expressions

A tuple literal is a sequence of element expressions of the appropriate type, separated by commas, enclosed in parentheses. For example, (1, One) is an (Int, Result) expression.

Other than literals, the only tuple expressions are symbols that are bound to tuple values, array elements of tuple arrays, and callable invocations that return tuples.

## User-Defined Type Expressions

A literal of a user-defined type consists of the type name followed by a tuple literal of the type’s base tuple type. For example, if IntPair is a user-defined type based on (Int, Int), then IntPair(2, 3) is a valid literal of that type.

Other than literals, the only expressions of a user-defined type are symbols that are bound to values of that type, array elements of arrays of that type, and callable invocations that return that type.

## Unwrap Expressions

In Q#, the unwrap operator is a trailing exclamation mark !. For example, if IntPair is a user-defined type with the underlying type (Int, Int) and s is a variable with value IntPair(2, 3), then s! is (2, 3).

For user-defined types defined in terms of other user-defined types, you can repeat the unwrap operator. For example, s!! indicates the doubly-unwrapped value of s. Thus, if WrappedPair is a user-defined type with underlying type IntPair, and t is a variable with value WrappedPair(IntPair(1,2)), then t!! is (1,2).

The ! operator has higher precedence than all other operators other than [] for array indexing and slicing. ! and [] bind positionally; that is, a[i]! is read as ((a[i])!): take the ith element of a, unwrap it, and then get the 3rd element of the unwrapped value (which must be an array).

The precedence of the ! operator has one impact that might not be obvious. If a function or operation returns a value that then gets unwrapped, the function or operation call must be enclosed in parentheses so that the argument tuple binds to the call rather than to the unwrap. For example:

let f = (Foo(arg))!;    // Calls Foo(arg), then unwraps the result
let g = Foo(arg)!;      // Syntax error


## Array Expressions

An array literal is a sequence of one or more element expressions, separated by commas, enclosed in square brackets []. All elements must be compatible with the same type.

Given two arrays of the same type, use the binary + operator to form a new array that is the concatenation of the two arrays. For example, [1,2,3] + [4,5,6] = [1,2,3,4,5,6].

### Array Creation

Given a type and an Int expression, use the new operator to allocate a new array of the given size. For example, new Int[i + 1] allocates a new Int array with i + 1 elements.

Empty array literals, such as [], are not allowed. Instead, you can create an array of length zero by using new T, where T is a placeholder for a suitable type.

The elements of a new array initialize to a type-dependent default value. In most cases, this is some variation of zero.

For qubits and callables, which are references to entities, there is no reasonable default value. Thus, for these types, the default is an invalid reference that you cannot use without causing a runtime error, similar to a null reference in languages such as C# or Java. Arrays containing qubits or callables must be initialized with non-default values before you can use their elements safely. For suitable initialization routines, see Microsoft.Quantum.Arrays.

The default values for each type are:

Type Default
Int 0
BigInt 0L
Double 0.0
Bool false
String ""
Qubit invalid qubit
Pauli PauliI
Result Zero
Range The empty range, 1..1..0
Callable invalid callable
Array['T] 'T

Tuple types initialize element-by-element.

### Array Elements

Given an array expression and an Int expression, form a new expression using the array element operator []. The new expression is the same type as the element type of the array. For example, if a is bound to an array of type Double, then a is a Double expression.

If the array expression is not a simple identifier, you must enclose it in parentheses to select an element. For example, if a and b are both arrays of type Int, an element from the concatenation is expressed as:

(a + b)


All arrays in Q# are zero-based. That is, the first element of an array a is always a.

### Array Slices

Given an array expression and a Range expression, form a new expression using the array slice operator [ ]. The new expression is the same type as the array and contains the array items indexed by the elements of the Range, in the order defined by the Range. For example, if a is bound to an array of type Double, then a[3..-1..0] is a Double[] expression that contains the first four elements of a but in the reverse order as they appear in a.

If the Range is empty, then the resulting array slice is zero length.

Just as with referencing array elements, if the array expression is not a simple identifier, you must enclose it in parentheses to slice it. For example, if a and b are both arrays of type Int, a slice from the concatenation is expressed as:

(a+b)[1..2..7]


#### Inferred start/end values

Starting with our 0.8 release, we support contextual expressions for range slicing. In particular, you may omit range start and end values in the context of a range slicing expression. In that case, the compiler applies the following rules to infer the intended delimiters for the range:

• If the range start value is omitted, then the inferred start value

• is zero if no step is specified or the specified step is positive.
• is the length of the sliced array minus one if the specified step is negative.
• If the range end value is omitted, then the inferred end value

• is the length of the sliced array minus one if no step is specified or the specified step is positive.
• is zero if the specified step is negative.

Some examples are:

let arr = [1,2,3,4,5,6];
let slice1  = arr[3...];      // slice1 is [4,5,6];
let slice2  = arr[0..2...];   // slice2 is [1,3,5];
let slice3  = arr[...2];      // slice3 is [1,2,3];
let slice4  = arr[...2..3];   // slice4 is [1,3];
let slice5  = arr[...2...];   // slice5 is [1,3,5];
let slice7  = arr[4..-2...];  // slice7 is [5,3,1];
let slice8  = arr[...-1..3];  // slice8 is [6,5,4];
let slice9  = arr[...-1...];  // slice9 is [6,5,4,3,2,1];
let slice10 = arr[...];       // slice10 is [1,2,3,4,5,6];


### Copy-and-Update Expressions

Since all Q# types are value types (with the qubits taking a somewhat special role), formally a "copy" is created when a value is bound to a symbol or when a symbol is rebound. That is to say, the behavior of Q# is the same as if a copy were created using an assignment operator.

Of course, in practice, only the relevant pieces are recreated as needed. This affects how you copy arrays because it is not possible to update array items. To modify an existing array requires leveraging a copy-and-update mechanism.

You can create a new array from an existing array via copy-and-update expressions, which use the operators w/ and <-. A copy-and-update expression is an expression of the form expression1 w/ expression2 <- expression3, where

• expression1 must be type T[] for some type T.
• expression2 defines which indices in the array specified in expression1 to modify. expression2 must be either type Int or type Range.
• expression3 is the value(s) used to update elements in expression1, based on the indices specified in expression2. If expression2 is type Int, expression3 must be type T. If expression2 is type Range, expression3 must be type T[].

For example, the copy-and-update expression arr w/ idx <- value constructs a new array with all elements set to the corresponding elements in arr, except for the element(s) specified by idx, which is set to the value(s) in value.

Given arr contains the array [0,1,2,3], then

• arr w/ 0 <- 10 is the array [10,1,2,3].
• arr w/ 2 <- 10 is the array [0,1,10,3].
• arr w/ 0..2..3 <- [10,12] is the array [10,1,12,3].

#### Copy-and-update expressions for named items

Similar expressions exist for named items in user-defined types.

For example, consider the type

newtype Complex = (Re : Double, Im : Double);


If c contains the value of type Complex(1., -1.), then c w/ Re <- 0. is an expression of type Complex that evaluates to Complex(0., -1.).

### Jagged Arrays

A jagged array, sometimes called an "array of arrays," is an array whose elements are arrays. The elements of a jagged array can be of different sizes. The following example shows how to declare and initialize a jagged array representing a multiplication table.

let N = 4;
mutable multiplicationTable = new Int[][N];
for (i in 1..N) {
mutable row = new Int[i];
for (j in 1..i) {
set row w/= j-1 <- i * j;
}
set multiplicationTable w/= i-1 <- row;
}


### Arrays of callables

You can also create an array of callables.

• If the common element type is an operation or function type, all of the elements must have the same input and output types.
• The element type of the array supports any functors that are supported by all of the elements. For example, if Op1, Op2, and Op3 all are Qubit[] => Unit operations, but Op1 supports Adjoint, Op2 supports Controlled, and Op3 supports both:
• [Op1, Op2] is an array of (Qubit[] => Unit) operations.
• [Op1, Op3] is an array of (Qubit[] => Unit is Adj) operations.
• [Op2, Op3] is an array of (Qubit[] => Unit is Ctl) operations.

However, while the operations (Qubit[] => Unit is Adj) and (Qubit[] => Unit is Ctl) have the common base type of (Qubit[] => Unit), arrays of these operations do not share a common base type.

For example, [[Op1], [Op2]] would currently raise an error because it attempts to create an array of the two incompatible array types (Qubit[] => Unit is Adj)[] and (Qubit[] => Unit is Ctl)[].

## Conditional Expressions

Given two expressions of the same type and a Boolean expression, form a conditional expression using the question mark, ?, and the vertical bar |. Given a==b ? c | d, the value of the conditional expression is c if a==b is true and d if it is false.

### Conditional expressions with callables

Conditional expressions may evaluate to operations that have the same inputs and outputs but support different functors. In this case, the type of the conditional expression is an operation with inputs and outputs that support any functors supported by both expressions. For example, if Op1, Op2, and Op3 all are Qubit[]=>Unit, but Op1 supports Adjoint, Op2 supports Controlled, and Op3 supports both:

• flag ? Op1 | Op2 is a (Qubit[] => Unit) operation.
• flag ? Op1 | Op3 is a (Qubit[] => Unit is Adj) operation.
• flag ? Op2 | Op3 is a (Qubit[] => Unit is Ctl) operation.

If either of the two possible result expressions include a function or operation call, that call only takes place if that result is the one that is the value of the call. For example, in the case a==b ? C(qs) | D(qs), if a==b is true, then the C operation is invoked, and if it is false then only the D operation is invoked. This approach is similar to short-circuiting in other languages.

## Callable Expressions

A callable literal is the name of an operation or function defined in the compilation scope. For example, X is an operation literal that refers to the standard library X operation, and Message is a function literal that refers to the standard library Message function.

If an operation supports the Adjoint functor, then Adjoint op is an operation expression. Similarly, if the operation supports the Controlled functor, then Controlled op is an operation expression. For more information about the types of these expressions, see Calling operation specializations.

Functors (Adjoint and Controlled) bind more closely than all other operators, except for the unwrap operator ! and array indexing with [ ]. Thus, the following are all valid, assuming that the operations support the functors used:

Adjoint Op(qs)
Controlled Op(controls, targets)


### Type-parameterized callable expressions

You can use a callable literal as a value, for example, to assign it to a variable or pass it to another callable. In this case, if the callable has type parameters, you must provide the parameters as part of the callable value.

A callable value cannot have any unspecified type parameters. For example, if Fun is a function with the signature 'T1->Unit:

let f = Fun<Int>;            // f is (Int->Unit).
let g = Fun;                 // This causes a compilation error.
SomeOtherFun(Fun<Double>);   // A (Double->Unit) is passed to SomeOtherFun.
SomeOtherFun(Fun);           // This also causes a compilation error.


## Callable Invocation Expressions

Given a callable expression (operation or function) and a tuple expression of the input type of the callable's signature, you can form an invocation expression by appending the tuple expression to the callable expression. The type of the invocation expression is the output type of the callable's signature.

For example, if Op is an operation with the signature ((Int, Qubit) => Double), Op(3, qubit1) is an expression of type Double. Similarly, if Sin is a function with the signature (Double -> Double), Sin(0.1) is an expression of type Double. Finally, if Builder is a function with the signature (Int -> (Int -> Int)), then Builder(3) is a function from Int to Int.

Invoking the result of a callable-valued expression requires an extra pair of parentheses around the callable expression. Thus, to invoke the result of calling Builder from the previous paragraph, the correct syntax is:

(Builder(3))(2)


When invoking a type-parameterized callable, you can specify the actual type parameters within angle brackets < > after the callable expression. This action is usually unnecessary as the Q# compiler infers the actual types. However, it is required for partial application if a type-parameterized argument is left unspecified. It is also useful when passing operations with different functor supports to a callable.

For example, if Func has signature ('T1, 'T2, 'T1) -> 'T2, Op1 and Op2 have signature (Qubit[] => Unit is Adj), and Op3 has signature (Qubit[] => Unit), to invoke Func with Op1 as the first argument, Op2 as the second, and Op3 as the third:

let combinedOp = Func<(Qubit[] => Unit), (Qubit[] => Unit is Adj)>(Op1, Op2, Op3);


The type specification is required because Op3 and Op1 have different types, so the compiler will treat this as ambiguous without the specification.

## Operator Precedence

• All binary operators are right-associative, except for ^.

• Brackets, [ ], for array slicing and indexing, bind before any operator.

• The functors Adjoint and Controlled bind after array indexing but before all other operators.

• Parentheses for operation and function invocation also bind before any operator but after array indexing and functors.

Q# operators in order of precedence, from highest to lowest:

Operator Arity Description Operand Types
trailing ! Unary Unwrap Any user-defined type
-, ~~~, not Unary Numeric negative, bitwise complement, logical negation Int, BigInt or Double for -, Int or BigInt for ~~~, Bool for not
^ Binary Integer power Int or BigInt for the base, Int for the exponent
/, *, % Binary Division, multiplication, integer modulus Int, BigInt or Double for / and *, Int or BigInt for %
+, - Binary Addition or string and array concatenation, subtraction Int, BigInt or Double, additionally String or any array type for +
<<<, >>> Binary Left shift, right shift Int or BigInt
<, <=, >, >= Binary Less-than, less-than-or-equal, greater-than, greater-than-or-equal comparisons Int, BigInt or Double
==, != Binary equal, not-equal comparisons any primitive type
&&& Binary Bitwise AND Int or BigInt
^^^ Binary Bitwise XOR Int or BigInt
||| Binary Bitwise OR Int or BigInt
and Binary Logical AND Bool
or Binary Logical OR Bool
.. Binary/Ternary Range operator Int
? | Ternary Conditional Bool for the left-hand-side
w/ <- Ternary Copy-and-update See Copy-and-update expressions

## Next steps

Now that you can work with expressions in Q#, move on to Operations and Functions in Q# to learn how to define and call operations and functions.