Two's complement

As explained in Building Quantum Software with Python

Two's complement is a signed binary representation in which the leading bit distinguishes nonnegative values from negative values and an m-bit string represents values from −2^(m−1) through 2^(m−1)−1.

Two's complement represents signed integers in a fixed-width binary register. The leading bit distinguishes nonnegative values from negative values: an m-bit string covers −2^(m−1) through 2^(m−1)−1. section 10.1.5

In quantum function encoding, two's complement provides a practical interpretation for value qubits:

• Strings beginning with 0 represent nonnegative outputs. section 10.1.5
• Strings beginning with 1 represent negative outputs. section 10.1.5
• With four value qubits, the representable range is −8 through 7. section 11.1
• A grid-state visualization can display negative values by setting neg=True. section 10.1.5

Encoding f(k) = k² − 3

For inputs 0 through 3, the lowest output is −3 and the highest is 0. The output −3 is encoded as 1101 in a four-bit value register; the function's grid shape remains the same but shifts to include negative values. section 10.1.5

Figure 10.10 The quantum state encoding the function f(k) = k² – 3 for 0 ≤ k < 4

View the quantum-state figure for f(k) = k² − 3 and its two's complement output encoding. section 10.1.5

How Building Quantum Software with Python covers it

How deep it goes

Defines sign-bit ranges and demonstrates four-bit encodings, but does not derive two's complement arithmetic.

Examples

Uses f(k)=k²−3 with output 1101 for −3 and f(k)=k²−5 with a four-qubit range from −8 through 7.

What it assumes

Assumes binary input variables and quantum registers while supplying the signed interpretation needed for negative outputs.

What you can do after

Apply it via build_polynomial_circuit, grid_state, and oracle_match_0 to encode and search signed quantum function outputs.

How it connects

Connects binary variables, quantum states, value registers, function encoding, oracle construction, and Grover adaptive search.

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Also explained in

• 

  liveBookmanning.com
  Modern CTwo's complement representation encodes negative integers by complementing the corresponding positive representation and adding one, with the high-order bit indicating negativity.
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  liveBookmanning.com
  Modern C, Third EditionTwo's complement is a signed-integer representation in which the high-order bit contributes negative weight and the remaining bits contribute positive weights.
• 

  liveBookmanning.com
  Clojure: The Essential ReferenceTwo's complement is a binary representation of signed integers in which negative values are encoded by complementing the bits of their magnitudes and adding one.

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Related concepts

• sign representationkind of · A sign representation is a scheme for encoding the sign and magnitude of a signed numerical value in bits.
• binary stringuses · A binary string is a finite, ordered sequence of binary digits, each having the value zero or one.
• integerimplements · An integer is a whole-number value with no fractional part, including negative numbers, zero, and positive numbers.

This explainer was generated by Manning's knowledge graph from the text of Building Quantum Software with Python. The section links point to the parts of the book it draws on.