Characterization of Quantum States Based on Creation Complexity

Abstract

The creation complexity of a quantum state is the minimum number of elementary gates required to create it from a basic initial state. The creation complexity of quantum states is closely related to the complexity of quantum circuits, which is crucial in developing efficient quantum algorithms that can outperform classical algorithms. A major question unanswered so far is what quantum states can be created with a number of elementary gates that scales polynomially with the number of qubits. In this work, it is first shown that for an entirely general quantum state it is exponentially hard (requires a number of steps that scales exponentially with the number of qubits) to determine if the creation complexity is polynomial. Then, it is shown that it is possible for a large class of quantum states with polynomial creation complexity to have common coefficient features such that, given any candidate quantum state, an efficient coefficient sampling procedure can be designed to determine if the state belongs to the class or not with arbitrarily high success probability. Consequently, partial knowledge of a quantum state's creation complexity is obtained, which can be useful for designing quantum circuits and algorithms involving such a state.

A quantum encryption design featuring confusion, diffusion, and mode of operation

“…Consequently the ability of Eve to efficiently determine the configuration of a polynomial state leads to her ability to tell if a general quantum state is polynomial or not. This is a contradiction to the result proved in our previous work [20] that it is exponentially hard to determine if a general quantum state can be created within polynomial number of gates. We therefore conclude that the key of our quantum encryption scheme is secure even if Eve has gained significant information on the ciphertext state.…”

“…Secondly, even if Eve is able to read the ciphertext—assuming the rare and can-be-avoided scenario that Alice sends the same ciphertext state many times and Eve is able to gain statistical knowledge of it—it is still highly difficult for her to deduce the key or the plaintext from the ciphertext. The detail of this reasoning is presented in the Supplementary Information S1 where the quantum state complexity theory in our previous study has been used 25 . Furthermore, this compromising scenario of Alice sending the same copy of the ciphertext many times can be totally avoided by the confusion, diffusion, and mode of operation to be introduced in the following sections.…”

“…To understand Statement 1 we first cite the result from our previous study on state complexity [20] that any sequence of 1-qubit and 2-qubit elementary gates is equivalent to a sequence of 2-qubit controlled-unitary gates or   C U 's. All quantum states that can be created by polynomially long sequences of 1-qubit and 2-qubit elementary gates (or concisely all polynomial states) thus correspond to all the sequences of   i C U 's with the lengths smaller than k cn for some constant c and k such that k cn is overwhelmingly smaller than 2 n .…”

The Quantum Condition Space

“…In the meanwhile, the number P of quantum projection measurements needed is a constant in the qubit number n, thus as long as the initiation of the quantum state vector has a polynomial complexity of O(poly(n)), then the total complexity of quantum evaluation of the event probability will be polynomial. This is indeed possible for any quantum states that have polynomial complexity, such as those that have been characterized as the "standard states" in our previous work on quantum state complexity [27] [(…”

“…Shown in Equation ( 2) is a 4-qubit "minimal standard state" whose quantum state vector can be efficiently initiated by the procedure described in Ref. [27]. The coefficients a i , b i , c i are arbitrary complex numbers that satisfy…”