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.

Introduction

Gate-model quantum computers are realized by unitary operators (quantum gates) and quantum states[1-29]. As the technological limits of current semiconductor technologies will be reached within the next few years[30-40], fundamentally different solutions provided by quantum technologies will be significant in the experimental realization of future computations[15-18,31,32,41-72]. A quantum computer is set up with a quantum gate structure, that is, via a set of unitary operators. These quantum gates can realize different quantum operations and can be defined on different numbers of input quantum states[15-18,41-43,45,52,53]. In a quantum computer environment, the depth of the quantum gate structure refers to the number of time steps (time complexity) required for the quantum operations making up the circuit to run on the quantum hardware[15-18,41-43,45,52-59]. A crucial problem here is the time complexity reduction of the quantum gate structure of the quantum computer. Practically, this problem is such that an equivalent quantum state of the output quantum state of the original the reference quantum circuit (e.g., non-reduced time complexity circuit) can be obtained using a reduced time complexity quantum gate structure. Particularly, currently there exists no plausible and implementable solution for the time complexity reduction of quantum computers. Gate-model quantum computer implementations are affected by the problem of high time complexities and a universal (i.e., platform independent) and tractable solution for the time complexity reduction is essential. Relevant implication of this problem is the high economic cost of the physical apparatuses required for experimentally implementing practical quantum computation: specifically, the high economic cost of the high-precision quantum hardware elements required in the implementation of high-performance quantum circuits.

The quantum circuit of the quantum computer is modeled as an arbitrary quantum circuit with an arbitrary circuit depth formulated via a unitary sequence of L unitary operators. Each unitary is set via a particular Pauli operator and gate parameter (see also Section 2 for a detailed description). The input problem fed into the quantum computer is an arbitrary computational problem with an objective function C. The C objective function is a subject of maximization via the quantum computer, i.e., via the unitaries of the circuit structure of the quantum computer. The objective function can model arbitrary combinatorial optimization problems[9,10,42], large-scale programming problems[10] such as the graph coloring problem, molecular conformation problem, job-shop scheduling problem, manufacturing cell formation problem, or the vehicle routing problem[10]. For a detailed description of input problems, we suggest[2-4,8-10,42-45].

Another important application of gate-model quantum computations is the near-term quantum devices of the quantum Internet[20,30,36-39,46-49,59,61,62,73-93].

Here, we define a quantum algorithm for the time complexity reduction of any quantum circuit of quantum computers set up with an arbitrary number of unitary gates. The aim of the proposed framework is to reduce the time complexity of an arbitrary reference quantum circuit and a maximization of the objective function of the computational problem fed into the quantum computer. The method defines the reduced time complexity equivalent of the reference quantum circuit and recovers the reference output quantum state via the reduced time complexity quantum circuit (Note: the terminology of quantum state refers to an input or output quantum system, while the terminology of quantum gate refers to a unitary operator.). The reduced structures are determined via a pre-processing phase in the logical layer, and only the reduced time complexity quantum circuit and reduced quantum state are implemented in the physical layer. The pre-rocessing phase integrates a machine learning[94-97] unit for the parameter optimization. The high complexity reference quantum circuit and reference quantum input are characterized only in the pre-processing phase without any physical level implementation. The framework applies a quantum algorithm on the output of the reduced quantum gate structure to recover the equivalent quantum state of the output quantum state of the non-reduced reference structure. In particular, the proposed framework and the defined quantum algorithm are universal since they have no requirements for the structure of the reference (e.g., non-reduced) quantum circuit subject to be reduced, for the number of unitaries in the reference structure, for the size of the input quantum state of the reference quantum circuit, nor for the dimensions of the actual quantum state. The quantum algorithm is defined as a fixed, auxiliary hardware component for an arbitrary quantum computer environment, with a pre-determined constant computational complexity as an auxiliary cost of the application of the algorithm. Specifically, we prove that the auxiliary cost of the proposed quantum algorithm is orders lower than the reachable amount of the reduction in time complexity, and the computational cost of the quantum algorithm becomes negligible in practice. The method also allows significantly reducing the economic cost of physical layer implementations, since the required elements and high-cost hardware components can be reduced in an experimental setting.

The novel contributions of our manuscript are as follows:

We define a quantum algorithm for circuit depth reduction of quantum circuits of gate-model quantum computers.

We define the computational cost of the proposed quantum algorithm and prove that it is significantly lower than the gainable reduction in time complexity.

The algorithm provides a tractable solution to reduce circuit depth and the economic cost of implementing the physical layer quantum computer by reducing quantum hardware elements.

The results are useful for experimental gate-model quantum computations and near-term quantum devices of the quantum Internet.

This paper is organized as follows. Section 2 defines the system model. Section 3 proposes the quantum algorithm and proves the computational complexity. Section 4 discusses the performance of the algorithm. Finally, Section 5 concludes the results. Supplemental material is included in the Appendix.

System Model

Let QG0 be a reference quantum circuit (quantum gate structure) with a sequence of L unitaries[42], defined aswhere is the L-dimensional vector of the gate parameters of the unitaries (gate parameter vector),and an i-th unitary gate U(θ) is evaluated aswhere P is a generalized Pauli operator acting on a few quantum states (qubits in an experimental setting) formulated by the tensor product of Pauli operators {σ, σ, σ}[42]. Note, that in (1) identifies a unitary resulted from the serial application of the L unitary operators U(θ)U(θ) … U1(θ1), and for an input quantum state |φ〉,

A qubit system example with a sequence of L unitaries is as follows. Let assume that the QG0 structure of the quantum computer consists of g single-qubit and m two-qubit unitaries, L = g + m, such that a j-th single-qubit gate realizes an operator, while a two-qubit gate realizes a operator (see also[42]). Then, at a particular objective function C of an arbitrary computational problem subject of a maximization via the quantum computer, the sequence from (1) can be rewritten aswherewhere is the gate parameter vector of the g single-qubit unitaries,while B is defined aswithandwhere B = X, while the two-qubit unitaries are defined aswhere 〈jk〉 ∈ QG0 is a physical connection between qubits j and k in the hardware-level of the QG0 structure of the quantum computer, is the gate parameter vector of the m two-qubit unitarieswhilewhere C is a component of the objective function, while unitary U(C, γ) for a given 〈jk〉 is defined aswhere

At a particular physical connectivity of QG0, the objective function C therefore can be written aswhere C(z) is the objective function component evaluated for a given 〈jk〉, aswhile z is an N-length input bitstring,where z identifies an i-th bit, z ∈ {−1, 1}.

For a given z, a |z〉 computational basis state is defined asand the |ϕ〉 output system of QG0 is asthat can be evaluated further via (6) and (11), as

To achieve the quantum parallelism, the input system |φ〉 = |X〉 of the quantum computer is set as an N-length d-dimensional (d = 2 for a qubit system) quantum state in the superposition of all possible d states. For a qubit system, it means that input |X〉 is an N-qubit quantum state in a superposition of all possible 2 states |0〉 to |2 − 1〉, and the computations are performed on 2 states in parallel in the quantum computer.

Let |X〉 be a superposed input system of the non-reduced QG0 gate structure:where |x〉 is an i-th input state (represented as an N-length bit string for a qubit system), i = 1, …, n, n = d, of the QG0 structure of the quantum computer.

To describe the parallel processing of the n input vectors of |X〉 (see (22)), {|x1〉, …, |x〉} of |X〉 (see (22)) in the quantum computer, let be the gate parameter vector associated with a given |x〉 of |X〉:

Let X be the classical representation of |X〉 in (22) to getwhere x is the classical representation of |x〉. (Note, that X and x are not accessible in the quantum computer, since the quantum algorithm operates in the quantum regime on quantum states. The classical representation is used only as an abstracted auxiliary representation to describe the steps of the algorithm in a plausible manner).

Then, let be the non-reduced gate structure matrix of QG0:whereand is the unitary sequence associated with |x〉 in QG0, defined as

At an n-dimensional output vectorand the |Y〉 output quantum state of the non-reduced QG0 structure is

To define the reduced gate structure, QG*, it is necessary to find a reduced with a reduced input , for all i.

Then, let be the classical representation of the reduced quantum state fed into QG*, asandwhere N* is the number of d-dimensional (physical) quantum states that formulate , n* = d, while the unitaries of QG* arewhereand is the reduced unitary sequence associated with , defined as

The pre-processing phase determines output Z of QG* as a classical representationand the output quantum state |Z〉 of QG* therefore yields

The notations of the system model are also summarized in Table A.1 of the Supplemental Information.

Problem statement

Problems 1–3 summarize the problems to be solved.

Problem 1

Find a classical pre-processing for calculating the reduced input system and the gate parameters of the QG* reduced time complexity gate structure.

Problem 2

Find a universal (independent of the number L of the unitaries in QG0) unitary operator U with a set of quantum registers to recover output |Y〉 of the non-reduced QG0 structure from output |Z〉 of the QG* reduced time complexity gate structure.

Problem 3

Determine the time complexity of U and the reduction in the overall time complexity of QG*.

Theorems 1–3 give the resolutions of Problems 1–3.

The non-reduced time complexity quantum circuit QG0 (reference circuit) with an input quantum state |X〉 is showed in Fig. 1(a). Figure 1(b) depicts the system model for the problem resolution. The method is realized via unitary U and pre-processing, such that U is implemented in the physical layer, while is only a logical-layer process. Only the reduced input quantum state and the reduced quantum gate structure QG* must be built up in the physical layer to yield the reference output system |Y〉 of the reference circuit QG0 via |Y〉. In both cases, the output states are measured via a measurement M to get a classical bitring for the objective function evaluation. As a next step, the gate parameter values of the unitaries of the circuits are calibrated until an optimal objective function value is not reached. The calibration of the gate parameters is a separate optimization procedure and its aim is fundamentally differ from the aim of , and therefore it is not part of the circuit depth reduction method. Note that existing algorithms can be utilized for this task (such as a the algorithms proposed in[19] and[20], or some gradient independent methods[98]).

Figure 1

(a) The non-reduced time complexity quantum circuit QG0 (reference circuit) with an input quantum state |X〉. The output of QG0 is |Y〉. The state |Y〉 is measured via a measurement M to get the classical string z to evaluate the objective function C(z). (b) System model of the time complexity reduction scheme. Pre-processing phase : the Y classical representation of output |Y〉 of QG0 is pre-processed by the pre-processing unit . Unit contains a computational block that outputs a vector κ, fed into an machine learning control unit for the Δ error feedback. Unit outputs and the gate parameters of the reduced structure that defines QG*. Quantum phase: from and the gate parameters, and QG* are set up. System is fed into the reduced quantum circuit QG*. The output of QG* is |Z〉, which is fed into the U recovery quantum algorithm. The U quantum operation outputs the |Y〉 system, which is the reference output |Y〉 of the reference circuit QG0. The state |Y〉 is measured via a measurement M to get the classical string z to evaluate objective function C(z).

Pre-processing

Theorem 1

There exists a pre-processing to determine the input system and the gate parameters, i = , …, n, for the reduced QG* gate structure for an arbitrary non-reduced QG0 structure with and input |X〉.

Proof. The pre-processing phase can be decomposed as , where is a computational block, while is a machine learning control block to calibrate the results of . We first define block , then discuss . The pre-processing is a procedure to stabilize the output of the reduced quantum circuit. is defined between the components and to evaluate and to set the gate parameters of the reduced quantum circuit structure QG* using the reference output |Y〉 of QG0. Note, as the output |Y〉 is fed into an M measurement block, the measurement results provide a feedback to calibrate in every subroutine of the protocol to produce a final saturated output. The Δ output of the machine learning control unit is used as a feedback in unit . For the definition of Δ, see (116) in Algorithm 1.

In the computational block, the reduced and are determined for ∀i, in the following manner. Note, since outputs the parameters of the reduced quantum gate structure, the extra complexity of a quantum structure can be replaced with classical complexity in the form of machine learning in the proposed framework.

Operation sets a one-dimensional discrete cosine transform[99] in the reduction method, thus a matrix G is defined as a generator matrix to evaluate the output coefficients of , see later (45). The definition of (see later in (40)) comes from the fact that any U unitary operator can be rewritten via the cos and sin functions, and using cosine functions rather than sine functions is critical for a compression[99]. In our setting, this is because fewer cosine functions are needed to approximate a particular U unitary operator.

Let x be the classical representation of |x〉, and be the classical representation of |y〉. Using the sequences of the L unitaries in (29), define a matrix G with n coefficients a, i = 1, …, n, aswherewhere θ identifies the gate parameter of a j-th unitary U(θ) associated to an i-th input x, while unitary sequence to an i-th input x iswhere P is a generalized Pauli operator.

First, the operation (one-dimensional discrete cosine transform[99]) is applied to the input matrix G from (37),where c is the p-th coefficient of ,whereand A is

The coefficients of defines matrix γ aswhere · is the inner product,where coefficients a-s are given in (37), and χ iswhere ς is an n-length vector

Then, the n-length output vector κ of is defined aswhere Y is given in (28), while Ω is as

Then, using the coefficients (41), (42) and (43) of , of the reduced state from (31) can be evaluated via the components of of (30). A p-th input for QG* is defined via (49) asand the reduced quantum gate sequence of in QG*, aswhere P is a generalized Pauli operator, and is as

Therefore, the quantum state |Z〉 of QG* is

The description of the machine learning control unit is as follows. Unit uses the results of to provide feedback for the pre-processing via supervised machine learning control.

The machine learning algorithm for the pre-processing control is defined in Algorithm 1.

The steps of the pre-processing method is given in Procedure 1.

■

Quantum Algorithm

Theorem 2

The |Y〉 output of the non-reduced QG0 structure can be recovered from the output |Z〉 of the reduced structure QG* via a unitary operator U.

Proof. Let be the input quantum state fed into the reduced structure QG*, and let |Z〉 (see (53)) be the output of the reduced gate structure. The task here is therefore to recover from |Z〉. The problem is solved via a unitary U, as follows.

Without loss of generality, in an i-th step, i = 1, …, n, the goal of the U operation is to calculate the quantum state aswhere κ is as in (48), while ω = (ω, …, ω) is an n-length vector defined for a given j, aswhere ∑θ is as given inwhere is given in (52).

Then letsuch that

Applying U for all i, yields the recovered quantum state |Y〉 aswhere an i-th |x〉 is aswhere i ≤ n − 1, and p ≥ 0, and is as given in (50); while the gate parameters (see (39)) of the L unitaries for a given i are evaluated as

The unitary U is defined via a set of quantum registers aswhere |R〉 is the i-th quantum register. The registers are initialized via set aswhere κ is given in (48), while ∂ and η are initial parameters defined asandwherewhere , andwhere .

Then, unitary U is defined aswhereand U is a unitary defined aswith eigenstatewhere U0 is an initial unitary operator that prepares state |R5〉 = |ω〉 for a given index state |R4〉 = |i〉, where ω is given in (55); from an initial |R4〉|R5〉 = |i〉|0〉 asin the register set (see (63)), where is the CNOT operation, while is an oracle applied on to compute Φ (54), defined aswhere is the resulting register set, while is an oracle that outputs function f, as

Specifically, note that (70) changes only the phase of the state as , where f is given in (74), while

Applying (74) on (63) yields a register state aswhere is the eigenvalue of U in (70).

Then, using the register set (63), let |ϕ0〉 be the input state for U as

Applying (68) k-times on (77) yields

The k iteration number in (78) is a random number, k < c, where , and m is initialized as m = 1[99].

Then let O be an oracle defined on as

Applying OU0 on (78), outputs system state

In particular, in system state (80), the state of register |R6〉 istherefore yields (59), such thatholds for all i of |Y〉, due to the conditions set in the pre-processing procedure (see (67)).

Assuming that the input system (77) for U is prepared for R-times and the output register (81) is measured for R-times, i.e., U is applied for R times in overall, in an r-th repetition, r = 1, …, R, the parameters of the procedure can be valuated aswherewhere is the measured value of |Φ〉 in the r-th repetition of U, while q( is the number of coefficients have been already found[99].

The actual value of r requires no increment if the relationholds, where τ( is a threshold value in the r-th iteration. Otherwise, the value of r can be increased, r = r + 1, as r < R.

The steps of the quantum algorithm U are given in Algorithm 2.

Distortion measure

As (81) is prepared in Step 4 of U, the state |Y〉 can be measured to get the classical string z to evaluate objective function C(z), as follows. Measure register |R6〉 of via a measurement operator M to evaluate objective function C(z), where z is a classical string resulted from the measurement of |Y〉, while C is an objective function of an arbitrary computational problem fed into the quantum computer.

The distortion coefficient associated with the |Y〉 recovered quantum state (59) can be evaluated at a particular objective function C, associated to the computational problem fed into the quantum computer aswhere z is a classical string resulting from the M measurement of |Y〉, while z is a classical string resulting from the M measurement of |Y〉.

Precisely, assuming R measurement rounds, an average of distortion yieldswhere C((z) and C((z) are the objective function values respectively associated with z and z in the r-th round, r = 1, …, R.

Computational Complexity

Theorem 3

Quantum algorithm U can be implemented with time complexity for the time complexity reduction of any non-reduced QG0 with an arbitrary number of L unitaries.

Proof. Letbe a global space spanned by |i〉, an n-dimensional vector |b〉, and by |c〉, which represents the inner product state.

Particularly, the U unitary in (68) applied on an input |φ〉 formulated via set of quantum registers giveswherethus U can be interpreted as a rotation on an n-dimensional subspace , 0 ≤ i < n, i.e., on a span of all |i〉.

Let ∏ be the solution set with conditions (82) for all i of ∏,and let be the superposition of all solutions:

The operation U on |φ〉 yields the state (see (80)):thus, U is a rotation on the subspace by angle towards (92), aswhere |∏| is the number of solutions (cardinality of solution set ∏).

U can be implemented as a rotation of on subspace (instead of a rotation on global space (88)) via a generalized quantum searching[100] that yields time complexity for an arbitrarily large quantum circuit QG0. ■

Performance Evaluation

Assuming that the initial time complexity of the QG0 non-reduced gate structure iswhere N is the number of d-dimensional (physical) quantum states in the superposed input system, and L is the number of unitaries in QG0, the time complexity of the reduced QG* structure iswhere N* is the number of d-dimensional (physical) quantum states in the reduced superposed input system, and L* is the number of unitaries in the reduced gate structure QG*.

Since the complexity of the proposed scheme isthe result of (96) is a reduced time complexity with respect to (95), as the relationholds; thus

The overall complexity of the QG* reduced structure at the application of U is therefore

Figure 2 depicts the resulting time complexities for a qubit implementation (N-qubit superposed input system, and qubit gate structure with L unitaries).

Figure 2

The time complexities (number of operations) for an N-qubit system, d = 2, n = 2, with an initial non-reduced gate structure QG0 with L unitaries, L = {10, 100, 1000, 10000}. The time complexity of QG0 is , while is an upper a bound on of QG*, .

To achieve time complexity reduction using and QG* instead of |X〉 and QG0, the relation must be straightforwardly satisfied, i.e., the initial complexity has to be reduced by more than . Since the complexity of the procedure is independent from the actual size of the gate structure, the cost remains constant for an arbitrarily large L.

Conclusions

Gate-model quantum computers are equipped with a collection of quantum states and unitary quantum gates. The realization of the quantum circuit of a quantum computer requires high fidelity, high precision, and high-level control. Since both the timecomplexity (depth of the circuits) and the economic costs of these implementations are high in practical scenarios, a reduction of these costs is essential. Here, we defined a quantum algorithm for reducing the circuit depth of gate-model quantum computers. The method achieves a reduction in the physical layer allowing significantly reducing implementation costs. The framework is flexible and can be used for arbitrary circuit depths.

Submission note

Parts of this work were presented in conference proceedings[101].

Ethics statement

This work did not involve any active collection of human data.

Supplementary Information.