Circuit Depth Reduction for Gate-Model Quantum Computers

Quantum computers utilize the fundamentals of quantum mechanics to solve computational problems more efficiently than traditional computers. Gate-model quantum computers are fundamental to implement near-term quantum computer architectures and quantum devices. Here, a quantum algorithm is defined for the circuit depth reduction of gate-model quantum computers. The proposed solution evaluates the reduced time complexity equivalent of a reference quantum circuit. We prove the complexity of the quantum algorithm and the achievable reduction in circuit depth. The method provides a tractable solution to reduce the time complexity and physical layer costs of quantum computers.


A.1 Notations
The notations of the manuscript are summarized in Table A.1. QG * Reduced time complexity quantum gate structure of a quantum computer.

|X
Superposed input system of the non-reduced QG 0 gate structure, |X = where N is the number of the d-dimensional quantum states that formulate |X , and |x i is the i-th input vector, i = 1, . . . , n, n = d N . d Dimension of the quantum state.
N Number of d-dimensional quantum states in the input system.
n Number of vectors in the input system of N quantum states, n = d N .
L Number of unitary gates in the QG structure of the quantum computer.
where P i is a generalized Pauli operator formulated by a tensor product of Pauli operators σ x , σ y , σ z , while θ i is referred to as the gate parameter associated to U i (θ i ).

C
Classical objective function of a computational problem fed into the quantum computer.
C Computational block in the P pre-processing phase, outputs κ.
P Generalized Pauli operator formulated by the tensor product of Pauli operators σ x , σ y , σ z .

P
Pre-processing in the logical layer, P = C L , where C is a computational block, while L is a machine learning control block to calibrate the results of C .
L Machine learning control unit in the P pre-processing, outputs the ∆ error for feedback control.
Quantum algorithm for the recovery of reference output quantum state |Y of QG 0 from the output |Z of QG * .
x p A p-th input x p of the reduced structure QG * .

U( θ p )
Reduced quantum gate sequence of QG * .
Σθ p Sum of gate parameters in QG * .
H Hypothesis from L . |Φ i Quantum state, subject to be determined via U R as an inner product state |Φ i = |ω i · κ .
k Iteration number for the application of U R .
Parameters for the r-th repetition round of U R , r = 1, . . . , R, where R is the total number of repetitions.
Measured value of |Φ i in the r-th repetition of U R .
z R Classical string, results from the M measurement of |Y R .
z Classical string, results from the M measurement of |Y .
C (z R ) Objective function value evaluated via z R .

C (z)
Objective function value evaluated via z.
where C (r) (z) and C (r) (z R ) are the objective function values associated to z and z R in the r-th round, r = 1, . . . , R.
G {|i |b |c } A global space spanned by |i , an n-dimensional vector |b , and by |c that represents the inner product state.
|ϕ Input quantum state, formulated via the set R of quantum registers.

|ϒ
Superposition of all solutions.

O (NL)
Initial time complexity of the QG 0 non-reduced gate structure, where N is the number of d-dimensional (physical) quantum states in the superposed input system, and L is the number of unitaries in QG 0 .
O (N * L * ) Time complexity of the reduced QG * structure, where N * is the number of ddimensional (physical) quantum states in the reduced superposed input system, L * is the number of unitaries in the reduced gate structure QG * .
O ( √ n) Complexity of the proposed framework of U R (the P pre-processing is not implemented in the physical layer).