A method for piecewise addition into an accumulation register performed by a quantum computing system comprising one or more classical processors and quantum computing hardware comprising a plurality of qubits, the method using one or more carry runway registers, wherein the accumulation register comprises a first plurality of qubits of the quantum computing hardware with each qubit representing a respective bit of a first binary number, and wherein each carry runway register comprises multiple qubits of the quantum computing hardware representing a respective binary number, the method comprising: inserting, by the quantum computing system, the one or more carry runway registers into the accumulation register at respective predetermined qubit positions, respectively, of the accumulation register, the insertion comprising increasing the number of qubits of the accumulation register and designating sections of the lengthened register as carry runway registers such that the accumulation register is divided into at least a low part and a high part; initializing, by the quantum computing system, each qubit of each carry runway register in a plus state; applying, by the quantum computing system, one or more subtraction operations to the accumulation register, wherein each subtraction operation subtracts a state of a respective carry runway register from a portion of the accumulation register that begins at the predetermined qubit position where the respective initialized carry runway register was inserted and ends at a qubit position prior to a next inserted carry runway or a most significant bit of the first binary number, the subtraction comprising: subtracting a value encoded in the state of the carry runway register from a value of the state of the portion of the accumulation register to obtain a new value, and updating the state of the portion of the accumulation register to encode the new value; and adding, by the quantum computing system, one or more input binary numbers into the accumulation register using piecewise addition, the piecewise addition comprising: separating the input binary numbers into pieces corresponding to portions of the accumulation register in between each carry runway register, and performing a respective addition operation for each corresponding piece of an input binary number in parallel to add the piece of the input binary number to a state of a respective accumulation register portion, wherein carries of the addition operations are added to states of respective carry runway registers. inserting, by the quantum computing system, the one or more carry runway registers into the accumulation register at respective predetermined qubit positions, respectively, of the accumulation register, the insertion comprising increasing the number of qubits of the accumulation register and designating sections of the lengthened register as carry runway registers such that the accumulation register is divided into at least a low part and a high part; initializing, by the quantum computing system, each qubit of each carry runway register in a plus state; applying, by the quantum computing system, one or more subtraction operations to the accumulation register, wherein each subtraction operation subtracts a state of a respective carry runway register from a portion of the accumulation register that begins at the predetermined qubit position where the respective initialized carry runway register was inserted and ends at a qubit position prior to a next inserted carry runway or a most significant bit of the first binary number, the subtraction comprising: subtracting a value encoded in the state of the carry runway register from a value of the state of the portion of the accumulation register to obtain a new value, and updating the state of the portion of the accumulation register to encode the new value; and subtracting a value encoded in the state of the carry runway register from a value of the state of the portion of the accumulation register to obtain a new value, and updating the state of the portion of the accumulation register to encode the new value; and adding, by the quantum computing system, one or more input binary numbers into the accumulation register using piecewise addition, the piecewise addition comprising: separating the input binary numbers into pieces corresponding to portions of the accumulation register in between each carry runway register, and performing a respective addition operation for each corresponding piece of an input binary number in parallel to add the piece of the input binary number to a state of a respective accumulation register portion, wherein carries of the addition operations are added to states of respective carry runway registers. separating the input binary numbers into pieces corresponding to portions of the accumulation register in between each carry runway register, and performing a respective addition operation for each corresponding piece of an input binary number in parallel to add the piece of the input binary number to a state of a respective accumulation register portion, wherein carries of the addition operations are added to states of respective carry runway registers.