Efficient Algorithms for Optimal 4-Bit Reversible Logic System Synthesis

Abstract

Owing to the exponential nature of the memory and run-time complexity, many methods can only synthesize 3-bit reversible circuits and cannot synthesize 4-bit reversible circuits well. We mainly absorb the ideas of our 3-bit synthesis algorithms based on hash table and present the efficient algorithms which can construct almost all optimal 4-bit reversible logic circuits with many types of gates and at mini-length cost based on constructing the shortest coding and the specific topological compression; thus, the lossless compression ratio of the space of 𝑛-bit circuits reaches near 2×𝑛!. This paper presents the first work to create all 3120218828 optimal 4-bit reversible circuits with up to 8 gates for the CNT (Controlled-NOT gate, NOT gate, and Toffoli gate) library, and it can quickly achieve 16 steps through specific cascading created circuits.

1. Introduction

Quantum computer is equivalent to quantum Turing machine, and quantum Turing machine is equivalent to a quantum logic circuit. Therefore, the quantum computer can be constructed by cascading and combining the quantum logical gates. Nowadays, many kinds of reversible quantum gates have been proposed, for example, CNOT gate [1], Toffoli gate, and Fredkin gate [2]. How to automatically construct the quantum circuit at small cost using given quantum gates? Several approaches for reversible logic circuit synthesis have been presented. Song et al. [3] presented algebraic characteristics of reversible gates. Iwama et al. [4] introduced transformation rules for CNOT-based circuits. Miller et al. [5] gave a synthesis method based on truth table and used template technology to simplify the circuit. Mishchenko and Perkowski [6] proposed a Reed Muller-based algorithm for optimizing quantum circuit. Gupta et al. [7] also gave a heuristic algorithm based on Reed Muller; Li et al. [8] proposed a general template algorithm. Shende et al. [9, 10] reduced the synthesis for reversible logic circuit to permutation and gave an effective recursive algorithm. Then, Yang et al. [11] reduced the synthesis for reversible logic circuit to group theory and presented a novel algorithm based on group-theory algebraic software GAP, while its performance was better than most others. Until today, we have not found a general and effective algorithm for multivariable quantum circuit synthesis. Owing to the exponential nature of the memory and run-time complexity, many existing methods can only synthesize 3-bit reversible circuits, and they perform only four steps for the CNP library in 4-bit circuit synthesis with mini-length for memory overflow [11–13], however, [14–16] are able to achieve 12 steps by using an enhanced bidirectional synthesis approach. We mainly absorb the ideas of our efficient 3-bit synthesis algorithms based on hash table and present the novel and efficient algorithms which can construct almost all optimal 4-bit reversible logic circuits [17]. Using lossless compression and cascading created circuits, our algorithms can get all the mini-length circuits whose lengths range from 0 to 8 with numbers 1, 28, 576, 9886, 147841, 1986374, 24375385, 274500662, and 2819198076, respectively; it can synthesize all the optimal 16-gate mini-length circuits for the CNT library.

2. Background

Definition 1. Let 𝑀 be a finite set; then, a permutation of 𝑀 is a bijection 𝜎∶𝑀→𝑀. If |𝑀|=𝑚, 𝜎 is an 𝑚 permutation. Normally, the permutation of group theory begins from one, but our program in c++ can work well if the permutation contains zero. Let 𝑀={0,1,…,𝑚−1}, and a permutation 𝜎=_𝑝^{01⋯𝑚−1}_{0𝑝1⋯𝑝𝑚−1}=(𝑝₀,𝑝₁,…,𝑝_𝑚−1). It can be denoted as the cyclic form; for example, 𝜏=(𝑎₁𝑎₂⋯𝑎_𝑖), 𝑖⩽𝑚; that is, 𝑎₁↦𝑎₂⋯↦𝑎_𝑖↦𝑎₁, 𝜎⁻¹=^{𝑝0𝑝1⋯𝑝𝑚−1}_{01⋯𝑚−1}, and 𝜋_𝑒=^{01⋯𝑚−1}_{01⋯𝑚−1} is the identity permutation of 𝑀, such that 𝜋_𝑒∘𝜎=𝜎∘𝜋_𝑒=𝜎 for any permutation 𝜎 of 𝑀.

The reversible function can be described by permutation or truth table. The 3-bit reversible logic circuit in Figure 1 can be described by the permutation 𝜎=^01234567_26017354=(2,6,0,1,7,3,5,4)=(02)(1653)(47) or the truth table in Table 1, where the input or output is ⟨𝑥_𝑛−1,…,𝑥₁,𝑥₀⟩₂=∑^𝑛−1_𝑖=0(𝑥_𝑖⋅2^𝑖) and ⟨𝑦_𝑛−1,…,𝑦₁,𝑦₀⟩₂=∑^𝑛−1_𝑖=0(𝑦_𝑖⋅2^𝑖), respectively. Let the permutations of NOT gate, Toffoli gate, and CNOT gate be 𝜎₁, 𝜎₂, 𝜎₃; then, 𝜎=𝜎₁∘𝜎₂∘𝜎₃=𝜎₁𝜎₂𝜎₃.

Table 1 

True table.

Figure 1 

Quantum reversible logic circuit.

In order not to repeat explanation, the following definitions are given.(1)Let 𝐶 be any 𝑛-bit reversible logic circuit with length 𝑙; it is a cascade of quantum gates 𝑔₁,𝑔₂,…,𝑔_𝑙, denoted by 𝐶=𝑔₁𝑔₂⋯𝑔_𝑙.(2)Let 𝑆𝐹={1,2,…,𝑛!}, 𝑆𝑀={1,2,…,𝑚}, and 𝑆𝐵={0,1}.(3)For permutation 𝜎=(0,1,…,𝑛−1), there are 𝑛! kinds of permutations, which are denoted by 𝜎₁,𝜎₁,…,𝜎_𝑛!, respectively.(4)Let 𝜋(𝑔) be the permutation of quantum gate 𝑔; then, the permutation of circuit 𝐶 is 𝜋(𝐶)≡𝜋(𝑔₁𝑔₂⋯𝑔_𝑙)=𝜋(𝑔₁)𝜋(𝑔₂)⋯𝜋(𝑔_𝑙).(5)Let cost(𝐶) be the cost of quantum circuit 𝐶 and cost(𝑔) be the cost of quantum gate 𝑔.(6)All the circuits in the paper are the reversible logic circuits.

Because permutation is often used in our algorithms, the effective expression of permutation is very important to improve the algorithms. Thus, we present the shortest coding scheme with saving plenty of space, which maps a permutation to an integer with the fewest bytes.

Definition 2. Let 𝑂^𝑖_𝑝 be ordinal number of 𝑝_𝑖 in sequence 𝑝₀,𝑝₁,…,𝑝_𝑖,…,𝑝_2𝑛−1; then, it is the number of members which are both smaller than and before 𝑝_𝑖; that is, 𝑂^𝑖_𝑝=^𝑖−1_𝑗=0sgn𝑝_𝑖−𝑝_𝑗(𝑥)=(),,sgn1,𝑥>00,else. (1)

Ordinal numbers of all members in the sequence 𝑝₀,𝑝₁,…,𝑝_𝑖,…,𝑝_2𝑛−1 construct ordinal number sequence: (𝑂⁰_𝑝,𝑂¹_𝑝,…,𝑂^2𝑛_𝑝⁻¹), and obviously 𝑂⁰_𝑝≡0 and 𝑂^2𝑛_𝑝⁻¹≡𝑃₂^𝑛₋₁. For example, the ordinal number sequence of 𝜎=(2,6,0,1,7,3,5,4) is (0,1,0,1,4,3,4,4).

Lemma 3. The ordinal number of 𝑝_𝑖 in Definition 2 can also be defined as 𝑂^𝑖_𝑝=𝑝_𝑖−^2𝑛−1_𝑗=𝑖+1sgn𝑝_𝑖−𝑝_𝑗. (2)

Proof. From (1), 𝑂^𝑖_𝑝 is the number of members which are both smaller than and before 𝑝_𝑖, and ∑^2𝑛−1_𝑗=𝑖+1sgn(𝑝_𝑖−𝑝_𝑗) is the number of members which are both smaller than and after 𝑝_𝑖; so, the sum of them is the number of members which are smaller than 𝑝_𝑖 in the sequence 𝑂^𝑖_𝑝+∑^2𝑛−1_𝑗=𝑖+1sgn(𝑝_𝑖−𝑝_𝑗)=|{0,1,…,𝑝_𝑖−1}|=𝑝_𝑖.

Theorem 4. One can get the ordinal number sequence in Definition 2 by 2^2𝑛−2−2^𝑛−1 comparison times merely.

Proof. The comparison times using (1) and Lemma 3 are 𝑖 and 2^𝑛−1−𝑖 for computing 𝑂^𝑖_𝑝, respectively, and comparison times are all ∑^2𝑛−1_𝑖=1𝑖=∑^2𝑛−1_𝑖=1(2^𝑛−1−𝑖)=2^2𝑛−1−2^𝑛−1 for computing ordinal number sequence. These two formulas can be chosen depending on condition to reduce the comparisons times. We use (1) if 𝑖<2^𝑛−1−𝑖; that is, 𝑖≤2^𝑛−1−1, and use Lemma 3 if 𝑖>2^𝑛−1−𝑖; that is, 𝑖≥2^𝑛−1. So, our method is that using (1) when 𝑝_𝑖 is in the first half part of the sequence, and otherwise using Lemma 3, their comparison times are ∑^{2𝑛−1−1}_𝑖=1𝑖 and ∑^2𝑛−1_{𝑖=2𝑛−1}(2^𝑛−1−𝑖), respectively. Finally, the whole comparison times are ∑^{2𝑛−1−1}_𝑖=1𝑖+∑^2𝑛−1_{𝑖=2𝑛−1}(2^𝑛−1−𝑖)=2^2𝑛−2−2^𝑛−1; it is lesser than the half of 2^2𝑛−1−2^𝑛−1.

Definition 5. The shortest coding of the permutation (𝑝₀,𝑝₁,…,𝑝_2𝑛−1) is Code(𝑝₀,𝑝₁,…,𝑝_2𝑛−1)=∑^2𝑛−1_𝑖=1(𝑂^𝑖_𝑝⋅𝑖!).

Lemma 6. If the ordinal number sequences of two permutations are the same, then the two permutations must be also the same.

Proof. Suppose that we get any two permutations of the set 𝑆={0,1,…,2^𝑛−1}, 𝑃=(𝑝₀,𝑝₁,…,𝑝_2𝑛−1), and 𝑄=(𝑞₀,𝑞₁,…,𝑞_2𝑛−1); if their ordinal number sequences are all 𝐵=(𝑏₀,𝑏₁,…,𝑏₂^𝑛₋₁), then we will prove that 𝑃=𝑄; namely, for all 𝑖∈{0,1,…,2^𝑛−1}, 𝑝_{2𝑛−1−𝑖}=𝑞_{2𝑛−1−𝑖}.
Proof by Mathematical Induction
Basis. Clearly, 𝑂^2𝑛_𝑝⁻¹=𝑏₂^𝑛₋₁, 𝑂^2𝑛_𝑞⁻¹=𝑏₂^𝑛₋₁; then, 𝑃 has 𝑏₂^𝑛₋₁ elements which are lesser than 𝑝_2𝑛−1; that is, 𝑝_2𝑛−1 is (𝑏₂^𝑛₋₁+1)th small element in 𝑆. Similarly, 𝑞_2𝑛−1 is (𝑏₂^𝑛₋₁+1)th small element in 𝑆, and so 𝑝_2𝑛−1=𝑞_2𝑛−1=𝑏₂^𝑛₋₁; that is, 𝑝_{2𝑛−1−𝑖}=𝑞_{2𝑛−1−𝑖} for 𝑖=0.
Inductive Assumption. Let 𝑝_{2𝑛−1−𝑖}=𝑞_{2𝑛−1−𝑖} for 𝑖=0,1,…,𝑗.
Inductive Step. When 𝑖=𝑗+1, 𝑂^2𝑛_𝑝^−2−𝑗=𝑏₂^𝑛_−2−𝑗, and 𝑃 has 𝑏₂^𝑛_−2−𝑗 elements which are both lesser than and before 𝑝_{2𝑛−2−𝑗}; that is, 𝑝_{2𝑛−2−𝑗} is (𝑏₂^𝑛_−2−𝑗+1)th small element in {𝑝₀,𝑝₁,…,𝑝_{2𝑛−2−𝑗}}, and let 𝑇_𝑝={𝑝_{2𝑛−1−𝑗},𝑝_2𝑛−𝑗,…,𝑝_2𝑛−1}, 𝑇_𝑞={𝑞_{2𝑛−1−𝑗},𝑞_2𝑛−𝑗,…,𝑞_2𝑛−1}; then {𝑝₀,𝑝₁,…,𝑝_{2𝑛−2−𝑗}}=𝑆−𝑇_𝑝; that is, 𝑝_{2𝑛−2−𝑗} is (𝑏₂^𝑛_−1−𝑗+1)th small element in 𝑆−𝑇_𝑝. Similarly, 𝑞_{2𝑛−2−𝑗} is (𝑏₂^𝑛_−1−𝑗+1)th small element in 𝑆−𝑇_𝑞. Now, using the inductive assumption, we get 𝑇_𝑝=𝑇_𝑞; that is, 𝑆−𝑇_𝑝=𝑆−𝑇_𝑞, and so 𝑝_{2𝑛−2−𝑗}=𝑞_{2𝑛−2−𝑗}; thus, 𝑝_{2𝑛−1−𝑖}=𝑞_{2𝑛−1−𝑖} for 𝑖=𝑗+1.

Lemma 7. The coding function Code maps each ordinal number sequence to a distinct coding.

Proof. Let 𝑐=Code(𝑝₀,𝑝₁,…,𝑝_2𝑛−1)=∑^2𝑛−1_𝑖=1𝑂^𝑖_𝑝⋅𝑖!; that is, the function Code maps ordinal number sequence 𝐵=(𝑂⁰_𝑝,𝑂¹_𝑝,…,𝑂^2𝑛_𝑝⁻¹) to coding 𝑐, and so the remainder is 𝑂¹_𝑝 when 𝑐 is divided by 2, the quotient is ∑^2𝑛−1_𝑖=2(𝑂^𝑖_𝑝⋅𝑖!/2), and the remainder is 𝑂²_𝑝 when this quotient is divided by 3. Follow the same steps; the recursive formula is 𝑛₁=𝑐, 𝑛_𝑖+1=⌊𝑛_𝑖/(𝑖+1)⌋, 𝑂^𝑖_𝑝=𝑛_𝑖−(𝑖+1)𝑛_𝑖+1, 𝑖∈{1,2,…,2^𝑛−1}. Clearly, the coding function Code maps each ordinal number sequence to a distinct coding, and vice versa; so, it is a one-to-one correspondence between each coding and ordinal number sequence.

Theorem 8. The function Code is the shortest coding function.

Proof. 2^𝑛 different elements in {0,1,…,2^𝑛−1} totally have 2^𝑛! permutations. According to Lemma 6, two different ordinal number sequences corresponding to two permutations must be different, and according to Lemma 7, coding function Code maps each ordinal number sequence to a distinct coding. So, the codings of all permutations are different.
By Definition 5, max(𝑂^𝑖_𝑝)=𝑖, min(𝑂^𝑖_𝑝)=0, min(𝐻(𝑝₀,𝑝₁,…,𝑝_2𝑛−2,𝑝_2𝑛−1))=min(∑^2𝑛−1_𝑖=1(𝑂^𝑖_𝑝⋅𝑖!))=∑^2𝑛−1_𝑖=1(min(𝑂^𝑖_𝑝)⋅𝑖!)=∑^2𝑛−1_𝑖=1(0⋅𝑖!)=0, max(𝐻(𝑝₀,𝑝₁,…,𝑝_2𝑛−2,𝑝_2𝑛−1))=max(∑^2𝑛−1_𝑖=1(O^𝑖_𝑝⋅𝑖!))=∑^2𝑛−1_𝑖=1(max(𝑂^𝑖_𝑝)⋅𝑖!)=∑^2𝑛−1_𝑖=1(𝑖⋅𝑖!)=2^𝑛!−1. So, the number of all codings is 𝐿=MAX𝐻−MIN𝐻+1=2^𝑛!−1−0+1=2^𝑛!(𝑛>0). There are 2^𝑛 different elements in {0,1,…,2^𝑛−1} corresponding to 2^𝑛! different coding, and thus the minimum number of all the codings is 2^𝑛!. Therefore, the function Code is the shortest coding function.

The subject of topology is concerned with those features of geometry which remain unchanged after twisting, stretching, or other deformations of a geometrical space; any continuous change which can be continuously undone is allowed, but cannot be broken.

Definition 9. Line permutation is the operation which permutes quantum lines in quantum circuit without break. Obviously, it is a specific topological transformation. Let 𝑃_𝜎(𝑔) be quantum gate 𝑔 being operated by line permutation 𝜎. The circuit 𝐺=𝑔₁𝑔₂, operated by line permutation 𝜎, is 𝑃_𝜎(𝐺)=𝑃_𝜎(𝑔₁𝑔₂)=𝑃_𝜎(𝑔₁)𝑃_𝜎(𝑔₂).

For example, in Figure 2, quantum circuit 𝐺 is operated by line permutation 𝜎, where 𝜎=⁰¹²₂₀₁=(02)(01).

Figure 2 

The line permutation of quantum circuit.

The circuit after applying topological transformation can be simplified, because the gate's function cannot be changed when the order of its same kind of points is changed.

Consider 𝑃_𝜎(𝐶)=𝑃_𝜎(𝑔₁𝑔₂⋯𝑔_𝑙)=𝑃_𝜎(𝑔₁)𝑃_𝜎(𝑔₂)⋯𝑃_𝜎(𝑔_𝑙). If there are 𝑚 basal gates in quantum gate library 𝐿, 𝑔₁,𝑔₂,…, and 𝑔_𝑚, then 𝐿=⋃_{𝑗∈𝑆𝐹,𝑖∈𝑆𝑀}𝑃_{𝜎𝑗(𝑔𝑖)}, and the set of all permutations of the gates in 𝐿 is 𝜎=⋃_{𝑗∈𝑆𝐹,𝑖∈𝑆𝑀}𝑃_𝜎𝑗(𝜋(𝑔_𝑖)).

The set of all circuits of 𝑆 operated by line permutation 𝜎 is denoted by 𝑃_𝜎(𝑆)={𝑃_𝜎(𝐺)∣𝐺∈𝑆}.

Definition 10. Direction transform (Figure 3) has two kinds of operations, obverse direction transform and reverse direction transform. It is also a specific topological transformation; the obverse direction transform 𝑏 of gate 𝑔 is denoted by 𝑅_𝑏(𝑔). Thus, 𝑅₀(𝑔)=𝑔, 𝑅₁(𝑔)=𝑔⁻¹. Gate 𝑔 is symmetric, if 𝑔=𝑔⁻¹, and otherwise it is asymmetric. Let circuit 𝐺=𝑔₁𝑔₂, and the two gates are symmetric; then, 𝑅₁(𝐺)=𝑅₁(𝑔₁𝑔₂)=𝑔₂⁻¹𝑔₁⁻¹=𝑔₂𝑔₁.

Figure 3 

The direction transform of quantum circuit.

Consider 𝑅₁(𝐶)=𝑅₁(𝑔₁𝑔₂⋯𝑔_𝑙)=(𝑔₁𝑔₂⋯𝑔_𝑙)⁻¹=𝑔_𝑙⁻¹𝑔⁻¹_𝑙−1⋯𝑔₂⁻¹𝑔₁⁻¹. 𝑅₁(𝐶)=𝑔_𝑙𝑔_𝑙−1⋯𝑔₂𝑔₁, if all the gates in 𝐶 are symmetric. Direction transform can be used in our synthesis algorithms only if all quantum gates are symmetric or their inverse gates are all in the quantum gate library.

The set of all circuits of 𝑆 operated by direction transform 𝑏 is denoted by 𝑅_𝑏(𝑆)={𝑅_𝑏(𝐺)∣𝐺∈𝑆}.

From Definitions 9 and 10, the following properties can be gotten.

Property 1. 𝑃_𝜎(𝑃_𝜏(𝐶))=𝑃_𝜎∘𝜏(𝐶) and 𝑃_𝜎(𝑃_𝜎−1(𝐶))=𝑃_𝜎−1(𝑃_𝜎(𝐶))=𝐶, but 𝑃_𝜎(𝑃_𝜏(𝐶))≠𝑃_𝜏(𝑃_𝜎(𝐶)), where 𝑃_𝜎(𝑃_𝜏(𝐶)) denotes the circuit 𝐶 being operated by line permutation 𝜏 and line permutation 𝜎; so, 𝑃_𝜎(𝑃_𝜏(𝐶))=𝑃_𝜏∘𝜎(𝐶). 𝜎∘𝜎⁻¹=𝜎⁻¹∘𝜎=𝜋_𝑒, and thus 𝑃_𝜎−1(𝑃_𝜎(𝐶))=𝑃_𝜎(𝑃_𝜎−1(𝐶))=𝑃_𝜋𝑒(𝐶)=𝐶. Normally, 𝜏∘𝜎≠𝜎∘𝜏; hence, 𝑃_𝜎(𝑃_𝜏(𝐶))≠𝑃_𝜏(𝑃_𝜎(𝐶)).

Property 2. 𝑅_𝑏1(𝑅_𝑏2(𝐶))=𝑅_𝑏2(𝑅_𝑏1(𝐶))=𝑅_{𝑏1⊕𝑏2}(𝐶), where 𝑅_𝑏1(𝑅_𝑏2(𝐶)) denotes the circuit 𝐶 being operated by direction transform 𝑏1 and direction transform 𝑏2. The circuit is not changed, if 𝑏1=𝑏2, and otherwise, it is operated by reverse direction transform; thus, 𝑅_𝑏1(𝑅_𝑏2(𝐶))=𝑅_{𝑏1⊕𝑏2}(𝐶). In the same way, 𝑅_𝑏2(𝑅_𝑏1(𝐶))=𝑅_{𝑏1⊕𝑏2}(𝐶).

Property 3. 𝑅_𝑏(𝑃_𝜎(𝐶))=𝑃_𝜎(𝑅_𝑏(𝐶)), where 𝑅_𝑏(𝑃_𝜎(𝐶)) denotes the circuit 𝐶 being operated by line permutation 𝜎 and direction transform 𝑏, and 𝑃_𝜎(𝑅_𝑏(𝐶)) denotes that the circuit 𝐶 is operated by direction transform 𝑏 and line permutation 𝜎. Line permutation is vertical transform, and direction transform is horizontal transform; so, the order of the two operations does not affect their compound function, and thus 𝑅_𝑏(𝑃_𝜎(𝐶))=𝑃_𝜎(𝑅_𝑏(𝐶)).

We show by Properties 1, 2, and 3 that the order of the previous topological transformations except the line permutations in quantum circuits can not affect their compound functions.

Lemma 11. For all 𝑘∈𝑆𝐹, for all 𝑏∈𝑆𝐵, 𝑐𝑜𝑠𝑡(𝐶)=𝑐𝑜𝑠𝑡(𝑅_𝑏(𝑃_𝜎𝑘(𝐶))).

Proof. Consider any quantum gate 𝑔, operated by any line permutation 𝜎_𝑘, whose type is not changed so, cost(𝑃_𝜎𝑘(𝑔))=cost(𝑔). When any quantum circuit 𝐶 is operated by any direction transform, its direction may be changed, but its cost can not be changed; thus, for all 𝑏∈𝑆𝐵, cost(𝑅_𝑏(𝐶))=cost(𝐶), and then cost(𝑅_𝑏(𝑃_𝜎(𝐶)))=cost(𝑃_𝜎(𝐶))=∑^𝑙_𝑖=1cost(𝑃_𝜎(𝑔_𝑖))=∑^𝑙_𝑖=1cost(𝑔_𝑖)=cost(𝐶), where cost(𝐶) of optimal 𝐶 should be minimal with the cost function. If for all 𝑖∈{1,2,…,𝑙}, cost(𝑔_𝑖)=1, then cost(𝐶)=𝑙; that is, the length of 𝐶 is minimal, and thus minimum cost standard degenerates to mini-length standard.

Lemma 12. Any optimal circuit after any specific topological transformation must be optimal.

Proof. Let the circuit 𝐶 be optimal, and circuit 𝐷=𝑅_𝑏(𝑃_𝜎(𝐶)). Assume that circuit 𝐷 is not optimal, and so there is an optimal circuit 𝐸 with 𝜋(𝐸)=𝜋(𝐷) and cost(𝐸)<cost(𝐷); thus, 𝜋(𝑅_𝑏(𝑃_𝜎−1(𝐸)))=𝜋(𝑅_𝑏(𝑃_𝜎−1(𝐷)))=𝜋(𝑅_𝑏(𝑃_𝜎−1(𝑅_𝑏(𝑃_𝜎(𝐶)))))=𝜋(𝐶). Let circuit 𝐹=𝑅_𝑏(𝑃_𝜎−1(𝐸)); by the Lemma 11, there is 𝜋(𝐹)=𝜋(𝐶), and cost(𝐹)=cost(𝐸), cost(𝐷)=cost(𝐶). Thus, cost(𝐹)<cost(𝐶); that is, the functions of circuit 𝐹 and circuit 𝐶 are the same, but 𝐹 is more optimal than 𝐶 in contradiction to 𝐶 being optimal.

Lemma 13. By Definition 9, for all 𝑘∈𝑆𝐹, 𝑃_𝜎𝑘(𝐿)=𝐿.

Proof. By permutation group theory, for all 𝑖,𝑗∈𝑆𝐹, 𝜎_𝑖𝜎_𝑗∈{𝜎₁,𝜎₂,…,𝜎_𝑛!}, and 𝑖≠𝑗→𝜎_𝑖≠𝜎_𝑗, and thus for all 𝑘∈𝑆𝐹, 𝜎_𝑖𝜎_𝑘≠𝜎_𝑗𝜎_𝑘, |{𝜎₁𝜎_𝑘,𝜎₂𝜎_𝑘,…,𝜎_𝑛!𝜎_𝑘}|=𝑛!, {𝜎₁𝜎_𝑘,𝜎₂𝜎_𝑘,…,𝜎_𝑛!𝜎_𝑘}={𝜎₁,𝜎₂,…,𝜎_𝑛!}. Similarly, we have ∀𝑘∈𝑆𝐹,𝜎_𝑘𝜎₁,𝜎_𝑘𝜎₂,…,𝜎_𝑘𝜎_𝑛!=𝜎₁,𝜎₂,…,𝜎_𝑛!. (3) For all 𝑘∈𝑆𝐹, 𝐿_new=𝑃_𝜎𝑘(𝐿) = 𝑃_𝜎𝑘(⋃_{𝑗∈𝑆𝐹,𝑖∈𝑆𝑀}𝑃_𝜎𝑗(𝑔_𝑖)) = ⋃_{𝑗∈𝑆𝐹,𝑖∈𝑆𝑀}𝑃_{𝜎𝑗𝜎𝑘}(𝑔_𝑖)=⁽³⁾⋃_{𝑖∈𝑆𝑀}(⋃_{𝑗∈𝑆𝐹}𝑃_𝜎𝑗(𝑔_𝑖)) = ⋃_{𝑗∈𝑆𝐹,𝑖∈𝑆𝑀}𝑃_𝜎𝑗(𝑔_𝑖)=𝐿.

Definition 14. Let 𝐺 be a circuit, Min(𝜋(𝐺))=min{𝑅_𝑏(𝑃_𝜎𝑗(𝜋(𝐺)))∣𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}, and thus ∃𝑘∈𝑆𝐹, ∃𝑐∈𝑆𝐵, 𝑅_𝑐(𝑃_𝜎𝑘(𝜏))=Min(𝜋(𝐺)). Let Min(𝐺)=𝑅_𝑐(𝑃_𝜎𝑘(𝐺)); Min(𝐺) is the minimal permutation circuit of 𝐺. All circuits gotten by all the specific topological transformations of 𝐺 compose the set 𝑆=⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏(𝑃_𝜎𝑗(𝐺)). The function GetMin(𝜏,𝜎,𝑏) returns the minimal permutation 𝜋, line permutation 𝜎 and direction transform 𝑏, where 𝜋=𝑅_𝑏(𝑃_𝜎(𝜏))=Min(𝜏).

Theorem 15. From Definition 14, Min(𝐺) is a circuit of 𝑆, and 𝑆 can be gotten only by Min(𝐺); that is, the lossless compression ratio of the space is near 2×𝑛!.

Proof. Obviously, all the minimal permutation circuits of 𝑆 must be Min(𝐺),𝑆=⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏(𝑃_𝜎𝑗(𝐺)), Min(𝐺)=𝑅_𝑐(𝑃_𝜎𝑘(𝐺)). Let 𝑆^′ be the set of all circuits gotten by all the specific topological transformations of Min(𝐺); thus, 𝑆^′=⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏(𝑃_𝜎𝑗(Min(𝐺))) = ⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏(𝑃_𝜎𝑗(𝑅_𝑐(𝑃_𝜎𝑘(𝐺))))=^Property3⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏⊕𝑐(𝑃_{𝜎𝑘𝜎𝑗}(𝐺))=⁽³⁾⋃_{𝑗∈𝑆𝐹,𝑏∈𝑆𝐵}𝑅_𝑏(𝑃_𝜎𝑗(𝐺) = 𝑆, where |𝑆𝐹|=𝑛!, |𝑆𝐵|=2; that is, |𝑆𝐹|×|𝑆𝐵|=2×𝑛!, and few specific topological transformation circuits may be the same. Thus |𝑆|≤2×𝑛! and |𝑆|≈2×𝑛!. Then, the lossless compression ratio of the set 𝑆 is near 2×𝑛!.

Let 𝐿_𝑛,𝐺 be an 𝑛-bit quantum gate library with gates 𝐺, and 𝑇(𝐿_𝑛,𝐺) a set of all 𝑛-bit reversible circuits synthesized by any gates in 𝐿_𝑛,𝐺. For example, 2×𝑛!|_𝑛=4=48. The average compression ratio of 𝑇(𝐿_4,CNT), whose circuits are synthesized 8 layers at minimal cost, is 47.95.

Given two quantum logic circuits, 𝐷 and 𝐺, ∃ℎ,𝑗∈𝑆𝐹, ∃𝑏,𝑐∈𝑆𝐵, 𝑅_𝑏(𝑃_𝜎ℎ(𝐺))=𝑅_𝑐(𝑃_𝜎𝑗(𝐷))→𝐺,𝐷∈𝑆→Min(𝐷)=Min(𝐺).

Taking all specific topological transformations of a 3-bit circuit as an example, there are 2×𝑛!|_𝑛=3=12 kinds of transformations. In Table 2, the bidirectional arrow expresses exchanging the two quantum lines, and the minimal permutation circuit of all circuits is circuit 𝐵 by computing coding.

Table 2 

All the specific topological transformations of 3-bit logic circuit.

The way of computing the shortest coding of circuit 𝐴 in Table 2 is given. (1)By Definition 1, the permutation of circuit 𝐴 is 𝜎=(0,1,2,3,6,7,5,4).(2)By Definition 2, the ordinal number sequence of 𝜎 is (0,1,2,3,4,5,4,4). (3)By Definition 5, the shortest coding is 23759.

Definition 16. Given two circuit sets 𝑆 and 𝑄, the set of all circuits of 𝑆 cascaded by all circuits of 𝑄 is denoted by 𝑆×𝑄={𝐺₁𝐺₂∣𝐺₁∈𝑆∧𝐺₂∈𝑄}.

Lemma 17. Given two circuit sets 𝑆 and 𝑄, 𝑃_𝜎(𝑆×𝑄)=𝑃_𝜎(𝑆)×𝑝_𝜎(𝑄).

Proof. By Definition 16, 𝑆×𝑄={𝐺₁𝐺₂∣𝐺₁∈𝑆∧𝐺₂∈𝑄}, thus 𝑃_𝜎(𝑆×𝑄) = 𝑃_𝜎({𝐺₁𝐺₂∣𝐺₁∈𝑆∧𝐺₂∈𝑄}) = {𝑃_𝜎(𝐺₁𝐺₂)∣𝐺₁∈𝑆∧𝐺₂∈𝑄} = {𝑃_𝜎(𝐺₁)𝑃_𝜎(𝐺₂)∣𝑃_𝜎(𝐺₁)∈𝑃_𝜎(𝑆)∧𝑃_𝜎(𝐺₂)∈𝑃_𝜎(𝑄)} = 𝑃_𝜎(𝑆)×𝑃_𝜎(𝑄).

Theorem 18. Given the set 𝑆_𝑙 of all optimal circuits with length 𝑙, Min(𝑆_𝑙+1)=Min((⋃_{𝑏∈𝑆𝐵}𝑅_𝑏(Min(𝑆_𝑙)))×𝐿)=Min(𝑆_𝑙×𝐿).

For saving memory, we only store minimal permutation circuits Min(𝑆_𝑙). The common computing method is that, firstly, 𝑆_𝑙 can be gotten by decompressing Min(𝑆_𝑙), secondly, 𝑆_𝑙+1 can be gotten by 𝑆_𝑙×𝐿, and lastly, Min(𝑆_𝑙+1) can be gotten by compressing 𝑆_𝑙×𝐿. Its number of decompressing circuit is 2×𝑛!×|Min(𝑆_𝑙)|, but ours is zero based on Theorem 15. The number of cascading circuit and the number of compressing circuit in the common method all are 𝑛! times than ours; so, our method is better.

3. New Synthesis Algorithm

A quantum logic gate realizes certain permutation in essence; quantum circuit is the cascade of some quantum gates, and so the basic function of quantum circuit can be represented the multiplication of permutations. The basic idea of the hash-based 3-bit synthesis algorithm which we previously advanced is constructing an optimal circuit using WFS, making a one-to-one correspondence between the elements in the hash table and different circuits. Thus, we need only one step to check whether the circuit of the same function has already been found to decide whether it is optimal. But it is unfeasible if we use this algorithm directly in 4-bit circuit synthesis, since the length of the hash table is at least 𝑛²!; so, it is extremely memory consuming. Otherwise, if we use DFS, it will be meaningless that the algorithm runs too slowly or can not realize optimal. To sum up, we adopt the BFS, with higher speed and more synthesis capability, and some lossless data compression methods to reduce memory consumption.(1)Represent the permutation using the shortest encoding. Only ⌈log^2𝑛!₂|_𝑛=4⌉=45 bits is required to represent 4-bit circuit permutations using the permutation encoding method in Definition 5, rather than 64 bits as usual.(2)Compress the optimal circuits without loss. Using line permutation alone, nearly 𝑛!|_𝑛=4=24 times compression can be reached, while using both line permutation and direction transformation, nearly 𝑛!×2|_𝑛=4=48 times compression can be reached, which is the backbone of this paper.(3)Use hash table with length 2¹⁶ in the top of the data structure, with each element pointing to a different RB tree in Figure 4 and each node in the tree reduces 2 bytes. Hash table and RB tree always remain effective, and RB tree can also allocate the memory dynamically. Therefore the data structure can not only permit a faster access speed, but also save memory space.

Figure 4 

The data structure of the minimal permutation circuit with length 𝑖.

How to determine the length of hash table? Let the permutation of 𝑛-bit circuit be 𝜋, whose shortest coding is Code(Min(𝜋)), {0,1}^{⌈log2𝑛!⌉} in binary. Use the rear 𝑘×8 bits as the hash address, the rest ⌈log₂^𝑛!⌉−𝑘×8 bits as the value of the nodes in the RB tree. Suppose that the length of the elements in hash table is 𝑗 bytes; so, hash table uses 𝑗×2^8𝑘 bytes while the RB tree saves 𝑘×|𝑆[𝑖]| bytes, and the maximum space this structure saves is max𝑓𝑚(𝑘,𝑗,𝑖)=𝑘×|𝑆[𝑖]|−𝑗×2^8𝑘. Take the 4-bit circuit based on CNT quantum gate library as an example; each element in hash table is a pointer pointing to the different tree, taking 4 bytes. The number of minimum permutation circuits with length 8 is 58777916, when 𝑘=2, and the maximum of the function is 𝑓𝑚(2,4,8)=117293688; so, the optimal length of hash table is 2^8𝑘|_𝑘=2=65536.

In this paper, we use BFS to get all the first 𝑁 layers of optimal circuits. If the gate library is determined, then the optimal circuits are determined. In order to save time, we store all minimum permutation circuits in the first 𝑁 layers in a file. If CNT gate library is used, then 𝑁=8, and we get a 700 MB file. As there are considerable gates in the 4-bit circuit quantum gate library, thus lots of optimal circuits are generated at each length. When a circuit reaches a certain length, the memory will overflow, and so 𝑁 has practical upper lower limit related to memory. After the lossless compression mentioned earlier, only the minimum permutation circuits are stored, saving 47.95 times less of circuits. The length of the overall synthesis circuits reaches 8 instead of 6, 117.7 times larger; the synthesizable length grows form 12 to 16 [14].

3.1. Minimal Length Algorithm

The node type of our RB tree is defined as follows:

struct rbtnode {   gate; // the gate is in the end of the circuit 𝐺   cpm; // the permutation of the circuit Min(𝐺)   binv; // the current circuit is inverted or not   lnpm; // Min(𝜋(𝐺)) ≡ GetMin(𝜋(𝐺); binv; lnpm)   pcpm; // the permutation of the previous circuit   pbinv; // the previous circuit is inverted or not

}

To enhance readability, two ways are used to simplify the algorithms. (1)All permutations are not transformed into the relevant shortest codings. (2)The hash table in the top of Figure 4 is omitted, and all the minimum permutation circuits in each layer are only saved in a RB tree; for example, 𝑆[𝑖] is the RB tree which has saved all minimum permutation circuits with length 𝑖.

3.2. Algorithm for the Minimal Permutation Quantum Reversible Logic Circuits Representation in QML

For more details, see Algorithm 2.

3.3. General Algorithm for the Quantum Reversible Logic Circuits Representation in QML (Algorithm 1)

Input: Quantum Gate Library 𝐿 Output:  max𝑙,𝑁[0…max𝑙],𝑆[0…max𝑙] 1: 𝑆[0]={cmp∶𝜋𝑒},𝑗=0,𝑁[0]=1 2: while  𝑁[𝑗]≠0  do 3: 𝑗=𝑗+1,𝑆[𝑗]=∅ 4: for each node Node x in 𝑆[𝑗−1]  do 5:  𝑐=Node𝑥.cpm 6:  for  𝑣=0to1  do 7:   if  𝑣=1  then 8:    𝑐=𝑐−1 9:   end if 10:    for each gate 𝑔 in 𝐿  do 11:    𝑝=𝑐∘𝜋(𝑔),𝜋=GetMin(𝑝,𝜎,𝑏) 12:    if  𝜋∉⋃𝑗𝑖=0𝑆[𝑖]  then 13:     𝑆[𝑗]=𝑆[𝑗]∪gate∶𝑔,cpm∶𝜋,lnpm∶             𝜎,binv∶𝑏,pcpm∶𝑝,             pbinv∶𝑣 14:     if memory overflow error then 15:      free𝑆[𝑗],𝑗=𝑗−1,goto23 16:     end if 17:    end if 18:   end for 19:  end for 20: end for 21: 𝑁[𝑗]=∑𝐺∈𝑆[𝑗]⋃𝑖∈𝑆𝐹,𝑏∈𝑆𝐵𝑅𝑏(𝑃𝜎𝑖(𝐺)) 22: end while 23: max𝑙=𝑗

Algorithm 1 

Quantum_Minimum_Length (QML).

Input: Quantum Gate Library 𝐿, minimal permutation 𝑝,    circuit length 𝑙 Output: Circuit of minimal permutation 𝑝 with mini-length 1: compute 𝑆[0…max𝑙] as in QML(𝐿) for the first time; 2: 𝑖=𝑙,mynode[𝑖]=𝑠[𝑖].find(𝑝) 3: pcpm𝑥=mynode[𝑖].pcpm 4: while  𝑖>1  do 5: 𝑖=𝑖−1 6: mynode[𝑖]=𝑆[𝑖].find(pcpm𝑥) 7: pcpm𝑥=mynode[𝑖].pcpm 8: end while 9: 𝜎=mynode[1].lnpm,𝑏=mynode[1].binv 10: 𝐺1=𝑃𝜎𝑅𝑏(mynode[1].gate) 11: for  𝑖=2 to 𝑙  do 12: 𝜎=mynode[𝑖].lnpm,𝑏=mynode[𝑖].binv 13: 𝑐=mynode[𝑖].pbinv,𝑔=mynode[𝑖].gate 14: 𝐺𝑖=𝑅𝑏𝑃𝜎𝑅𝑐𝐺𝑖−1𝑔 15: end for 16: return  𝐺𝑙

Algorithm 2 

Minimal_Quantum_Representation (MQR).

For more details, see Algorithm 3.

Input: Quantum Gate Library 𝐿, permutation 𝑝 Output: Circuit of permutation 𝑝 with mini-length 1: if  ∃𝑙∈{0,1,2,…,max𝑙},GetMin(𝑝,𝜎,𝑏)∈𝑆[𝑙]  then 2: 𝐺=MQR(GetMin(𝑝,𝜎,𝑏),𝑙) 3: return  𝑅𝑏(𝑃𝜎−1(𝐺)) 4: else if   ∃𝑖∈𝑆𝐹,∃𝑏∈𝑆𝐵,∃𝑙∈{1,2,…,max𝑙},  ∃Node𝑥∈𝑆[𝑙],    GetMin(𝑅𝑏𝑃𝜎𝑖𝑝∘Node𝑥.cpm−1,𝜏,𝑐)     ∈𝑆[max𝑙]  then 5: 𝐺2=MQR(Node𝑥.cpm,𝑙) 6: 𝜌=GetMin𝑅𝑏𝑃𝜎𝑖(𝑃)∘Node𝑥.cpm−1,𝜏,𝑐 7: 𝐺1=MQR𝜌,max𝑙 8: return  𝑅𝑏𝑃𝜎𝑖−1𝑅𝑐𝑃𝜏−1𝐺1𝐺2 9: else 10: return  NULL 11: end if

Algorithm 3 

Quantum_Minimum _Representation (QMR).

4. Experimental Results

Our experiments were conducted using many benchmark functions for 4-bit reversible logic circuits synthesis, and [14, 15] dealt with the synthesis of 4-qubit circuit for the high complexity of the algorithm. Based on CNP quantum gate library, using the mini-length as criteria, [14] added 4 layers in the basis of [11] through bidirection synthesis, and another 4 layers by combining DFS, thus realized a 12 layers synthesis of arbitrary circuits. Yet [14] can only synthesize the first 4 layers of the optimal circuit at a time, while in our previous study, for example, the hash-based 3-bit synthesis algorithm, the average speed of the mini-length and minimum cost are 49.15 and 365.13 times of [11], respectively. In this paper, we used CNT quantum gate library and mini-length criteria, creating all optimal circuits with up to 8 gates. By bidirection cascading the generated circuits, we can quickly synthesize the optimal quantum circuits within the length of 16, without consuming more memory. Our bidirection cascading is quite different with the bidirection synthesis used in [14]; they calculated the head and tail of the circuit, respectively, then moved forward to the middle. In order to avoid repeated computation, we first synthesize the former parts of the circuit, then perform specific topology transformations on them and reuse them in the latter part.

To evaluate the ability of the algorithm while synthesizing complicated circuits, we have run our program on a great number of circuits, and none of them has been found not to be synthesized. Then, we only give two examples. (1) We cascaded the two optimal circuits 4_49 and Hwb4 to get one circuit in Figure 5A. By using the generated all minimal permutation circuits with up to 8 gates, it took only 35 s to generate circuit B. It is easy to prove that the permutation of both A and B are (15,2,3,12,5,9,1,11,0,10,14,6,4,8,7,13). (2) Synthesized Alhagi01 (2,12,8,13,0,9,6,15,10,11,14,4,5,3,1,7) circuit is given in Figure 6.

Figure 5 

The 4-bit reversible logic circuits synthesis.

Figure 6 

Alhagi01 circuit synthesis.

5. Conclusions

Based on the idea that the synthesis of reversible logic circuit is a permutation problem in essence, we present the novel and efficient quantum circuit synthesis algorithms. Among them, we elaborately construct a shortest encoding method of the permutation and compress the memory space of the 𝑛-bit optimal circuits to 2×𝑛! times less using certain topology transformation of quantum circuits. By bidirectional cascading of the generated optimal circuits, using several quantum gates and the mini-length cost metric, our algorithms can efficiently generate most optimal 4-bit reversible logic circuits.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 61070240, 61170321, 61272175, and 61103235), the Natural Science Foundation of College of Jiangsu Province (no. 10KJB520021), and Specialized Research Fund for the Doctoral Program of Higher Education (no. 20110092110024).