Extended superposed quantum-state initialization using disjoint prime implicants

Abstract

Extended superposed quantum-state initialization using disjoint prime implicants is an algorithm for generating quantum arrays for the purpose of initializing a desired quantum superposition. The quantum arrays generated by this algorithm almost always use fewer gates than other algorithms and in the worst case use the same number of gates. These improvements are achieved by allowing certain parts of the quantum superposition that cannot be initialized directly by the algorithm to be initialized using special circuits. This allows more terms in the quantum superposition to be initialized at the same time which decreases the number of gates required by the generated quantum array.

Binary superposed quantum decision diagrams

“…This shows that BSQDDs can be used to achieve an exponential reduction in the number of required gates over existing methods other than the ESQUID algorithm [9] for initializing some classes of quantum superpositions. However, the ESQUID algorithm [9] requires the quantum superposition to be represented using generalized phase groups; this is a significant drawback because finding a sequence of generalized phase groups that will result in an efficient quantum array is a difficult problem that is not solved by the ESQUID algorithm [9]. The ESQUID algorithm [9] also still requires more one and two qubit operations than quantum arrays generated using BSQDDs for a class of quantum superpositions even though the difference in complexity is not exponential.…”

“…BSQDDs have several important advantages over existing methods for synthesizing quantum arrays for initializing quantum superpositions. Some quantum superpositions can be initialized using only a linear number of one and two qubit operations with quantum arrays generated using BSQDDs while the Ventura-Martinez [11], SQUID [8] and Long-Sun [6] algorithms all require an exponential number of one and two qubit operations as is shown in Sect. 11.…”

“…However, the ESQUID algorithm [9] requires the quantum superposition to be represented using generalized phase groups; this is a significant drawback because finding a sequence of generalized phase groups that will result in an efficient quantum array is a difficult problem that is not solved by the ESQUID algorithm [9]. The ESQUID algorithm [9] also still requires more one and two qubit operations than quantum arrays generated using BSQDDs for a class of quantum superpositions even though the difference in complexity is not exponential. Additionally, the ESQUID algorithm is only capable of initializing a narrow class of quantum superpositions while BSQDDs can be used for any quantum superposition given an appropriate set of gates.…”

“…Another initialization algorithm that is based on a different idea is the Long-Sun algorithm [6] which requires no ancilla qubits but uses (n 2 2 n ) two qubit gates. The SQUID algorithm [8] is an improvement over the Ventura-Martinez algorithm [11] and uses O( pn) two qubit gates and requires n + 2 ancilla qubits where p is the number of disjoint phase groups in the phase map of the desired superposition. The ESQUID algorithm [9] is an extension of the SQUID algorithm [8] and uses O(bn) two qubit gates and requires n + 2 ancilla qubits where b is the number of disjoint generalized phase groups in the phase map of the desired superposition.…”

“…The SQUID algorithm [8] is an improvement over the Ventura-Martinez algorithm [11] and uses O( pn) two qubit gates and requires n + 2 ancilla qubits where p is the number of disjoint phase groups in the phase map of the desired superposition. The ESQUID algorithm [9] is an extension of the SQUID algorithm [8] and uses O(bn) two qubit gates and requires n + 2 ancilla qubits where b is the number of disjoint generalized phase groups in the phase map of the desired superposition. BSQDDs do not require the ancilla qubits or the special initialization operators used in the Ventura-Martinez [11], SQUID [8] and ESQUID [9] algorithms and can be used to find quantum arrays that are more efficient than those generated by the Ventura-Martinez [11], Long-Sun [6], SQUID [8] and ESQUID [9] algorithms.…”