Suppressing environmental noise in quantum computation through pulse control

A scheme based on pulse control is described for suppressing noise in quantum computation without the cost of stringent quantum computing resources. It is shown that environment-induced noise and decoherence, whether in quantum memory or in gate operations, all can be much reduced by applying a suitable sequence of bit-flipping and phase-flipping pulses. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.67.Hk; 03.67.Dd; 42.50.-p Quantum computation has become a very active field ever since the discovery that quantum computers can be much more powerful than their classical w x counterparts 1–3 . Quantum computers act as sophisticated quantum information processors, in which calculations are made by the controlled time evolution of a set of coupled two-level quantum systems Ž . qubits . Coherence in the evolution is essential for taking advantage of quantum parallelism. However, there is a major obstacle to the realization of quantum computation. Decoherence of the qubits caused by the inevitable interaction with noisy environment will make quantum information too fragile to be of any practical use. Recently, interest in quantum computation has increased dramatically because of two respects of advances toward overcoming the above ) Corresponding author. E-mail: [email protected] 1 E-mail: [email protected] difficulty. First, a combination of a series of innovative discoveries, such as quantum error correcting w x codes 4,5 , fault-tolerant error correction techniques w x w x 6 , and concatenated coding 7 , has yielded the w x important threshold result 7,8 , which promises that arbitrarily accurate quantum computation is possible provided that the error per operation is below a threshold value. Hence, noise below a certain level is not an obstacle to reliable quantum computation. Second, great progress has been made toward buildw x ing the necessary quantum hardwares 9–11 . Some simple but real quantum computations have been demonstrated in bulk spin resonance quantum sysw x tems 12,13 ; and a radical scheme, using semiconductor physics to manipulate nuclear spins, has been recently proposed, which indicates a promising route w x to large-scale quantum computation 14 . Fault-tolerant quantum error correction schemes are effective only when the error rate per operation is below a threshold value. Operations in quantum 0375-9601r99r$ see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 99 00592-7 ( ) L.-M. Duan, G.-C. GuorPhysics Letters A 261 1999 139–144 140 computers include transmission or storage of quantum states and quantum logic. For quantum logic, the y6 w x estimated threshold error rate is about 10 7,8 . With the present technology, the real noise seems necessarily beyond this threshold, even for the most w x promising quantum hardware 8 . Therefore, it is essentially important to further suppress noise before taking the procedure of fault-tolerant quantum error correction. Here, we propose a noise suppression scheme which operates by applying a sequence of bit-flipping and phase-flipping pulses on the system, with the pulse period less than the noise correlation time. It is shown that noise in the pulse-controlled system can be greatly reduced. The scheme based on pulse control is relatively easy to implement in experiments. In fact, clever Ž . pulse methods called refocusing techniques have been developed for years in nuclear magnetic resoŽ . nance spectroscopy NMR to effectively remove many kinds of interactions among the spins that are w x considered unwanted or uninteresting 15 . In a rew x cent work 16 , Viola and Lloyd used this technique to combat environmental noise in computer memory with a specific single-qubit dephasing model. Later, the technique was immediately extended to more w x general cases 17,18 , and a general theory for suppressing memory noise has been developed in the w x group context 19,20 . In this paper, we consider the general type of environmental noise, including classical or quantum dephasing and dissipation as its special cases. By applying suitable pulse sequences, it is found that the noise in the controlled system is much reduced; and furthermore, the scheme can be readily extended to suppress noise in quantum gates Žreferring to two-bit or multi-bit quantum gates, whose error rates are much larger than that of w x. single-qubit rotations 9,14,21 . Suppose p is the g error rate per operation, and p is the additional 0 error rate introduced by each pulse. Our result shows that the error rate per operation in the pulse-controlled system approximately reduces to p p t rt , g 0 dec c where t and t are respectively the decoherence dec c time and the noise correlation time. Since p is 0 normally very small, the reducing factor p t rt is 0 dec c less than 1 even that the noise correlation time is considerably smaller than the decoherence time. Therefore, by this scheme it is possible for the reduced error rate to attain the threshold value, say, 10. This is a desirous result for reliable quantum computation. The only requirement in our scheme is that the pulses should be applied frequently so that the pulse period is less than the noise correlation time. The scheme costs no additional quantum computing resources. First we show how to use pulse control to combat decoherence of a single qubit in quantum memory. The qubit is described by Pauli’s operator s . In the interaction picture, the most general form of the Hamiltonian describing single-qubit decoherence due Ž to environmental noise can be expressed as setting . "s1 a H t s s G t , 1 Ž . Ž . Ž . Ý I a asx , y , z where G t , generally dependent of time, are noise Ž . a terms, which may be classical stochastic variables or stochastic quantum operators, corresponding classical noise or quantum noise, respectively. For environmental noise in quantum computers, it is reasonable to assume that G t asx , y , z satisfy the Ž . Ž . a 2 : conditions G t s0 and Ž . a env G t G t s f j , 2 2 : Ž . Ž . Ž . Ž . a b a b env 2 : where PPP denotes average over the environenv Ž . ment. In Eq. 2 , all f j are correlation functions Ž . ab Ž . and js ty t rt . The quantity t characterizes the c c order of magnitude of the noise correlation times. Different noise correlation terms may have different correlation times. But we assume that they have the same order of magnitude, which is denoted by t . c For simplicity, in the following we directly call tc the noise correlation time. Suppose that the qubit is initially in a pure state Ž . : C 0 . Under the Hamiltonian 1 , after a short Ž . time t it evolves into a mixed state r t after taking Ž . average over the environment. The difference be: tween the states r t and C 0 stands for errors. Ž . Ž . The error rate p can be described by ps1yF t , Ž . where F t is the input-output state fidelity, defined Ž . Ž . 2 : as F t s C 0 r t C 0 . From Eqs. 1 and Ž . Ž . Ž . Ž . Ž . 2 , it is not difficult to obtain an explicit perturbative expression for the error rate. Up to the second order of the Hamiltonian, the result is t t t 2 1 a b 2 : ps2 Ds Ds f dt dt , 3 Ž . Ý HH s a b 2 1 ž / t 0 0 c a ,b ( ) L.-M. Duan, G.-C. GuorPhysics Letters A 261 1999 139–144 141 a a 2 a: 2 : where Ds ss y s , and PPP denotes avs s Ž . erage over the system. Eq. 3 is derived with a pure input state. However, it remains true when the qubit is initially in a mixed state. In this case, we define the error rate by ps1yF t , where F t is the Ž . Ž . e e w x entanglement fidelity 22 , a natural extension of the input-output state fidelity to the mixed state circumstance. With this definition, the expression for the Ž . error rate remains completely same as Eq. 3 . Now we show how to use pulse control to reduce the error rate in quantum memory. We apply a sequence of bit-flipping and phase-flipping pulses on the qubit, with the pulse period t and pulse width D t . It is required that t < t so that the environw w D ment-induced system evolution during the short time t is negligible. At time ts0, no pulse is applied, w and its operation is represented by the unit operator I. At time ts t , we begin to apply in turn D the bit-flipping and the phase-flipping pulses. The pulse-induced operations are thus respectively I,s ,s ,s ,s , PPP . Four pulse periods make up a control period. Let U t2 denote t1 X X t2 4 T exp yiH H t dt , where T PPP indicates Ž . 1⁄2 5 t I 1 that time-ordered product is taken in the bracket. In Ž . Ž the nq1 th control period form time 4nt to time D . 4 nq1 t , the effective system evolution under Ž . D pulse control is represented by the following evolution operator: Ž . z 4 nq4 t x Ž4 nq3. t D D U ss U s U Ž . nq1 4 nq3 t Ž4 nq2. t D D =s U Ž4 nq2. Ds U Ž4 nq1. tD Ž4 nq1. t 4 nt D D Ž . 4 nq4 tD X X z z syT exp yi s H t s dt Ž . H I 1⁄2 Ž . 4 nq3 tD Ž . 4 nq3 tD X X y y yi s H t s dt Ž . H I Ž . 4 nq2 tD Ž . 4 nq2 tD X X x x yi s H t s dt Ž . H I Ž . 4 nq1 tD Ž . 4 nq1 tD X X yi H t dt Ž . H I 5 4ntD Ž . 4 nq1 tD syT exp yiH 1⁄2 4ntD X X X a s G t dt . 4 Ž . Ž . Ý a 5 asx , y , z The effective noise terms G X 4nt q t with 0F tŽ . a D Ž . 4 t in Eq. 4 are defined as follows: D 3 1 t X G 4nt q t s h G 4nt q jt q , 5 Ž . Ž . Ý a D a j a D D ž / 4 4 js0 where the 3=4 coefficient matrix 1 1 y1 y1 w x hs h s . 6 Ž . 1 y1 1 y1 a j 1 y1 y1 1 After pulse control, the only difference in the system Ž . evolution 4 is that the noise terms G t are Ž . a replaced by the corresponding effective noise terms X Ž . Ž . G t . Form Eqs. 2 and 5 , it is not difficult to get Ž . a the correlations of the effective noise G X t . Then, Ž . a Ž . substituting these correlations into Eq. 3 , we obtain the error rate p after pulse control. The final result c is 2 p t c D sa , 7 Ž . ž / p tc where t is the noise correlation time, and a is a c Ž dimensionless near-to-1 factor its explicit form can w x. be found in 17 , which is unimportant to our result and will be omitted in the following discussion. Ž . From Eq. 7 , it follows that through pulse control the error rate in quantum memory can be reduced by a factor proportional to the second order of the rate of the pulse period to the noise correlation time. In fact, by applying more complicate pulse sequences, the error rate can be further reduced. For example, we may apply the pulse sequence I,s ,s ,s , I,s ,s , s , I, PPP , with the pulse period t . In this sequence, a control period consists of D eight pulse periods. We can similarly calculate the error rate of the system controlled by this pulse sequence. The result is that the error rate is reduced by a factor proportional to t rt . In general, if we Ž . D c apply a pulse sequence with the control period consisting of 2 nq1 pulse periods, the error rate is able to be reduced by a factor proportional to t rt 2 . Ž . D c In the above, we considered decoherence of a single qubit. Now, suppose that there are L qubits, described respectively by Pauli’s operators s . In l the interaction picture, the Hamiltonian describing decoherence of L qubits can be generally expressed ( ) L.-M. Duan, G.-C. GuorPhysics Letters A 261 1999 139–144 142