In some example embodiments a logical block comprising twelve inputs and two six-input lookup tables (LUTs) is provided, wherein four of the twelve inputs are provided as inputs to both of the six-input lookup tables. This configuration supports efficient field programmable gate array (FPGA) implementation of multipliers. Each six-input LUT comprises two five-input lookup tables (LUT5s) that are used to form Booth encoding multiplier building blocks. The five inputs to each LUT5 are two bits from a multiplier and three Booth-encoded bits from a multiplicand. By assembling building blocks, multipliers of arbitrary size may be formed.

A programmable apparatus for executing a function is disclosed. The programmable apparatus includes a physical interface configured to be connected with an external apparatus. The programmable apparatus also includes a function logic circuit configured to execute the function on the programmable apparatus. The programmable apparatus further includes a plurality of peripheral logic circuits, each of which is configured to connect the function logic circuit with the physical interface using a respective protocol. The programmable apparatus also includes a selector circuit configured to select one from among the plurality of the peripheral logic circuits to activate.

In some example embodiments a logical block comprising twelve inputs and two six-input lookup tables (LUTs) is provided, wherein four of the twelve inputs are provided as inputs to both of the six-input lookup tables. This configuration supports efficient field programmable gate array (FPGA) implementation of multipliers. Each six-input LUT comprises two five-input lookup tables (LUT5s) that are used to form Booth encoding multiplier building blocks. The five inputs to each LUT5 are two bits from a multiplier and three Booth-encoded bits from a multiplicand. By assembling building blocks, multipliers of arbitrary size may be formed.

Methods and apparatus for piecewise addition into an accumulation register using one or more carry runway registers, where the accumulation register includes a first plurality of qubits with each qubit representing a respective bit of a first binary number and where each carry runway register includes multiple qubits representing a respective binary number. In one aspect, a method includes inserting the one or more carry runway registers into the accumulation register at respective predetermined qubit positions, respectively, of the accumulation register; initializing each qubit of each carry runway register in a plus state; applying one or more subtraction operations to the accumulation register, where each subtraction operation subtracts a state of a respective carry runway register from a corresponding portion of the accumulation register; and adding one or more input binary numbers into the accumulation register using piecewise addition.

A programmable apparatus for executing a function is disclosed. The programmable apparatus includes a physical interface configured to be connected with an external apparatus. The programmable apparatus also includes a function logic circuit configured to execute the function on the programmable apparatus. The programmable apparatus further includes a plurality of peripheral logic circuits, each of which is configured to connect the function logic circuit with the physical interface using a respective protocol. The programmable apparatus also includes a selector circuit configured to select one from among the plurality of the peripheral logic circuits to activate.

A programmable apparatus for executing a function is disclosed. The programmable apparatus includes a physical interface configured to be connected with an external apparatus. The programmable apparatus also includes a function logic circuit configured to execute the function on the programmable apparatus. The programmable apparatus further includes a plurality of peripheral logic circuits, each of which is configured to connect the function logic circuit with the physical interface using a respective protocol. The programmable apparatus also includes a selector circuit configured to select one from among the plurality of the peripheral logic circuits to activate.

In some example embodiments a logical block comprising twelve inputs and two six-input lookup tables (LUTs) is provided, wherein four of the twelve inputs are provided as inputs to both of the six-input lookup tables. This configuration supports efficient field programmable gate array (FPGA) implementation of multipliers. Each six-input LUT comprises two five-input lookup tables (LUT5s) that are used to form Booth encoding multiplier building blocks. The five inputs to each LUT5 are two bits from a multiplier and three Booth-encoded bits from a multiplicand. By assembling building blocks, multipliers of arbitrary size may be formed.

Described herein is a generalized optimal reduction scheme for reducing an array modulo a constant. The constant modulo operation calculates a result for array of bits x.sub.i, width n modulo an odd positive integer constant d, (e.g., x[n:0] mod d). Circuitry to perform such operation can be configured to compress the array of bits x.sub.i, width n into an array of bits y.sub.i width m. The techniques described herein enable the design of optimal circuitry via iterative exploration of all potential reduction strategies that are available given the input constraints.

Methods and apparatus for performing surface code computations using Auto-CCZ states. In one aspect, a method for implementing a delayed choice CZ operation on a first and second data qubit using a quantum computer includes: preparing a first and second routing qubit in a magic state; interacting the first data qubit with the first routing qubit and the second data qubit with the second routing qubit using a first and second CNOT operation, where the first and second data qubits act as controls for the CNOT operations; if a received first classical bit represents an off state: applying a first and second Hadamard gate to the first and second routing qubit; measuring the first and second routing qubit using Z basis measurements to obtain a second and third classical bit; and performing classically controlled fixup operations on the first and second data qubit using the second and third classical bits.

A programmable apparatus for executing a function is disclosed. The programmable apparatus includes a physical interface configured to be connected with an external apparatus. The programmable apparatus also includes a function logic circuit configured to execute the function on the programmable apparatus. The programmable apparatus further includes a plurality of peripheral logic circuits, each of which is configured to connect the function logic circuit with the physical interface using a respective protocol. The programmable apparatus also includes a selector circuit configured to select one from among the plurality of the peripheral logic circuits to activate.