Universal fault-tolerant quantum computation using fault-tolerant conversion schemes

In this paper, we present the fault-tolerant conversion between quantum Reed–Muller (QRM)(2, 5) and QRM(2, 7), and also the conversion between QBCH(15, 7) and QRM(2, 7). Either of the two schemes provides a method to realize universal fault-tolerant quantum computation. In particular, the gate overhead and logical error rate of a logical T gate are provided, as well as the comparison with magic state distillation scheme. In addition, we propose two other fault-tolerant conversion schemes based on ( u ∣ u + v ) and ( a + x ∣ b + x ∣ a + b − x ) constructions.


Introduction
A fault-tolerant computation architecture can remarkably allow arbitrarily accurate quantum computation against decoherence, provided that the error introduced by noise is below a certain threshold value [1,2]. The simplest fault-tolerant operation is the use of transversal gates [3]. Various stabilizer codes can provide transversal implementations for various gates [2], for example, the seven-qubit Steane code can realize the entire Clifford group transversally, while the 15-qubit quantum Reed-Muller (QRM) code admits a transversal = p ( ) T diag 1, e i 4 gate. An important universal gate set is the Clifford + T set. Unfortunately, it has been proved that no quantum code can be constructed to allow a universal set of transversal gates [4,5]. Therefore, other fault-tolerant methods are required.
Magic state distillation [6][7][8] is now a leading proposal to complete a universal gate set. The key idea is to accept many low-fidelity magic states prepared by state injection or the previous round of distillation, and then to filter them into high-fidelity ones. In that case, magic state distillation can be orders of magnitude more costly than the transversal implementation [9,10]. Another idea for universal fault-tolerant quantum computation is to combine the transversal gates in two different codes using fault-tolerant conversion scheme [11,12]. Under the subsystem stabilizer formalism [13], this conversion scheme can be interpreted as different gauge fixing procedures [14][15][16] on the same subsystem code.
In this work, we first present the fault-tolerant conversion between QRM(2, 5) and QRM (2,7) and the faulttolerant conversion between QBCH (15,7) and QRM (2,7). Both schemes enable us to circumvent the no-go on transversal gates in universal quantum computation. In particular, we give the gate overhead and logical error rate of a logical T gate using conversion scheme. For comparison, the results of using magic state distillation scheme are also provided. In addition, we present two other fault-tolerant conversion schemes based on the + ( | ) u u v construction and the + + + -( | | ) a x b x a b x construction. With a wide range of applications, these schemes are helpful in exploiting the advantages of different codes for specific scenarios [17,18].
The rest of this paper is organized as follows. In section 2, we describe the constructions and transversal properties of QRM codes and QBCH codes. In sections 3.1 and 3.2, we present the conversion between QRM (2,5) and QRM (2,7) and the conversion between QBCH (15,7) and QRM (2,7). The fault-tolerant logical T implementations are provided accordingly. In section 3.3, we illustrate two other conversion schemes based on the + ( | ) u u v construction and + + + -( | | ) a x b x a b x construction respectively. In section 4, we compare the gate overhead and logical error rate of a logical T gate using fault-tolerant conversion scheme and magicstate distillation scheme. Finally, we give concluding remarks in section 5.

Preliminaries
This section describes the classical and quantum error-correcting codes that are involved in the fault-tolerant code conversion procedures which we will discuss later. For the basic facts and notions in classical coding theories, the readers are referred to [2,19].
Shortened-RM codes and BCH codes.-Suppose that γ is a primitive (2 m −1)th-root of unity for m3. Let where wt(s) indicates the number of 1s in the binary expansion of s. The dual code( is not the dual of ( ) r m RM , . Furthermore, the generator matrix of where 1 is a vector of all 1s (the columns are permuted for later convenience). In particular, the generator matrix of e.g.
A BCH code BCH(d, m) of length 2 m −1 and designed distance d is a cyclic code of minimum distance at least d with a generator polynomial When  m 5 and   - and contains its dual [20].  [21]. Especially when  + m r 3 1, QRM(r, m) has a transversal T gate [11,22,23].
QBCH codes.-For  m 5 and   - In other words, given an indicating that c has weight divisible by 4. In that case, QBCH(d,7) allows the transversal implementation of a logical S gate on the first encoded qubit.

Fault-tolerant conversion schemes
In this section, we first present the conversion between QRM codes, and the conversion between QRM codes and QBCH codes. Based on the + ( | ) u u v construction and the + + + -( | | ) a x b x a b x construction, we then provide two other conversion schemes which are applicable to a wide range of quantum codes. qubits. Hence by using the fault-tolerant implementations of state preparations and stabilizer measurements described in [24], we can complete the fault-tolerant conversion between QRM(r, m) and QRM(r, m+1) as follows.
To convert from ( ) r m QRM , to QRM(r, m+1), we first fault-tolerantly prepare the 2 m -qubit entangled state Fñ | m and append it to the system. After fault-tolerantly measuring the stabilizer generatorś Ä -- of QRM(r, m+1), we remove any −1 syndrome bit revealed by a generator A using the so-called 'pure error' operator [11] Î á Ä Ä ñ Ä - except A. Finally, we perform a fault-tolerant error-correction operation through faulttolerant measurements of To convert from QRM(r, m+1) to QRM(r, m), we first fault-tolerantly measure , and restore any −1 syndrome bit using the associated 'pure error' operator in á´Ä ñ -- . After measuring the stabilizer generators 1 fault-tolerantly, we apply a recovery procedure to correct any single-qubit error. By discarding the additional 2 m qubits, we can finally obtain an encoded state of QRM(r, m).
Under the subsystem stabilizer formalism [13], the conversion schemes illustrated above can be recast as different gauge fixing procedures on the - . The first scenario (showed in figure 1 (a)) can be implemented by measuring all Z gauge operators and fixing the gauge qubit with the associated X gauge operator if the outcome is −1. The second scenario (showed in figure 1(b)) can be realized by measuring all X gauge operators and fixing the gauge qubit with corresponding Z gauge operator if the outcome is −1.
a The notation [[n, k, r, d]] denotes a subsystem code that has length n and minimum distance d with k logical qubits and r gauge qubits. b The gauge operators indicate the logical operators associated with gauge qubits.   (2, 7) consists in fixing all the 21 gauge qubits in encoded ñ |0 or +ñ | . Similar with the process in figure 1(b), the fault-tolerant conversion from QBCH (15,7) to QRM(2, 7) can be realized by first fault-tolerantly measuring the Z gauge operators and fixing the gauge qubit with corresponding X gauge operator if the outcome is −1. After fault-tolerant measurements of stabilizer generators H X 3,7 and H Z 3,7 , any single-qubit errors can be corrected with a recovery operation. By reversing these steps, the fault-tolerant conversion from QRM(2, 7) to QBCH (15,7) can be accomplished.

Other conversion schemes
We can tell that the key to a fault-tolerant conversion between two quantum codes is to correspond them to the same subsystem code. For example, the correspondence between QRM codes relies mainly on the recursive definition of shortened-RM codes (equation (3)). Following that idea, techniques of generating new codes from the old ones may also help to build the required correspondence. In this subsection, we consider the use of two well-known construction methods-the + ( | ) u u v construction and the + + + -    code CSS(C ⊥ , C ⊥ ). Assume that ÎĈ 1 1 and ÎC 1 1 , then we can choose a basis for the code space of  C such that logical X and Z operators on the first encoded qubit are transversal X and Z, respectively. Provided that all the other logical qubits are prepared in ñ |0 , the stabilizer generators for this encoded state  añ Î | C C are listed in if the outcome is −1. After fault-tolerant measurements of stabilizer generators   , fix the gauge qubit with corresponding gauge operator in á´- 2 if the outcome is −1, and perform a fault-tolerant error-correction operation. By discarding the additional n qubits, we can then obtain the required state of  1 . x b x a b x construction Let C 1 , C 2 be defined as above, then we can construct a [3n, 2k 1 +k 2 , min{2d 1 , a x b x a b x a b x

The
The dual codeD is a It is certain that if both C 1 and C 2 are self-orthogonal, D is also self-orthogonal.  are listed in  table 7. Clearly, the generators of bñ | D from the first 9 rows can be changed into without altering the code state. Hence we can correspond bñ subsystem code in table 8. In that case, the conversion between  1 and  D consists in fixing one half of the gauge qubits in encoded ñ |0 and the other half in encoded +ñ | . Note that the fault-tolerant conversion schemes based on the + ( | ) u u v construction and the x b x a b x construction only require that C 1 and C 2 are self-orthogonal codes satisfying Ì C C 2 1 , ÎĈ 1 1 and ÎC 1 1 . Hence, a wide range of codes could be applied to these schemes. Specifically by picking C 1 and C 2 to be Hamming codes, the Golay code derived from a variant of the x b x a b x construction [25], has shown advantages in constructing quantum codes with higher threshold [26]. In that case, the fault-tolerant conversion from quantum Hamming code to quantum Golay code can help us exploit the particular advantage of the latter code.

Performance analysis
In this section, we analyze the gate overhead and logical error rate of a logical T gate using fault-tolerant conversion scheme. For simplicity, we assume that the error probability of any individual component in the Table 7. Stabilizer generators for bñ

Stabilizer generators for bñ
conversion circuit are the same p, including qubits, gates and measurements. In addition, all physical gates are assumed to have equal costs.  [24], we list the gate overhead of these fault-tolerant circuits in table 9. In addition, we give the gate overhead of fault-tolerant conversion between QRM(2, 5) and QRM(2, 7) in tables 10 and 11. By adding 127 T gates, we can then obtain the gate overhead of a fault-tolerant logical T on QRM(2, 5) in table 12. It is clear that the gate overhead declines with a decreasing physical error probability p, especially when p falls from 5×10 −3 to 10 −3 . In addition, the fault-tolerant conversion from QRM (2,5) to QRM(2, 7) consumes much more physical gates than its reverse when p=5×10 −3 . The main reason is that fault-tolerant Fñ | 5 and Fñ | 6 preparations in the former case require intensive resources to maintain a reasonable probability of acceptance.
We give the gate overhead of fault-tolerant conversion between QBCH (15,7) and QRM (2,7) in tables 13 and 14. Combined with 127 T gates, the gate overhead of a fault-tolerant logical T gate on QBCH (15,7) is illustrated in table 15. By comparison with table 12 we can tell that, a fault-tolerant logical T on QBCH (15,7) consumes less physical gates than on QRM (2,5). This is natural since the ancilla state preparations are not required for QBCH (15,7) before the measurement of gauge operators. Also, the gate overhead drops sharply with p descending from 5×10 −3 to 10 −3 .

Comparison with magic state distillation scheme
For comparison, we investigate the gate overhead of using magic state distillation scheme [24,27]. A magic state +ñ | T can first be prepared by state injection and then be processed with several rounds of distillation procedures From the comparison between tables 21-23 we can tell that, given the same physical error probability p, the logical error rate of a logical T gate using magic state distillation scheme is in general much smaller than that of using conversion scheme. In fact, the logical T gate using conversion scheme performs better than the individual physical gates only when p<10 −6 , indicating a hard requirement on physical gate fidelity. The magic state distillation scheme, however, has a significant advantage in this aspect.

Conclusion
In this paper, we present the fault-tolerant conversion scheme between QRM(2, 5) and QRM (2,7), as well as the conversion between QBCH(15, 7) and QRM (2,7). Either of the two schemes enables us to fault-tolerantly perform a logical T gate without magic state distillation. The gate overhead is provided, along with the comparison with magic state distillation scheme. For QBCH (15,7), the conversion scheme can achieve a smaller overhead due to the lack of fault-tolerant state preparations. For QRM (2,5), the gate overhead of using magic state distillation scheme declines fast with a descending p and becomes smaller than using the conversion scheme when p 10 5 . In terms of the logical error rate, the distillation scheme has a significant advantage and enables a logical T gate to outperform the physical gate when p 10 4 . On the other hand, the key to converting between two different codes is to correspond them to the same subsystem code with different gauge qubits. In other words, classical codes that construct these two quantum codes need to be closely related and have common codewords. Under such circumstances, we consider the use of the + ( | ) u u v construction and the + + + -( | | ) a x b x a b x construction, and present two fault-tolerant conversion schemes between quantum codes derived from the new codes and old codes. These schemes have a wide range of applications and can thus contribute to exploiting the advantages of different codes for specific scenarios. We should note that all the proposed fault-tolerant conversion schemes could be well-generalized to higher-dimensional systems, at least to the p-ary systems. In that case, more methods of constructing new codes from old ones [28] could be investigated.