Algorithmic cooling

The algorithm operates on an array of equally (and independently) biased qubits. After the algorithm transfers heat (and entropy) from some qubits to the others, the resulting qubits are rearranged in increasing order of bias. Then this array is divided into two sub-arrays: "cold" qubits (with bias exceeding a certain threshold chosen by the user) and "hot" qubits (with bias lower than that threshold). Only the "cold" qubits are used for further quantum computation. The basic procedure is called "Basic Compression Subroutine"^[2] or "3 Bit Compression".^[14]

The reversible case can be demonstrated on 3 qubits, using the probabilistic approach. Each qubit is represented by a "coin" (two-level system) whose sides are labeled 0 and 1, and with a certain bias: each coin is independently with bias ${\displaystyle \varepsilon }$, meaning probability ${\displaystyle {\frac {1+\varepsilon }{2}}}$ for tossing 0. The coins are ${\displaystyle A,B,C}$ and the goal is to use coins ${\displaystyle B,C}$ to cool coin (qubit) ${\displaystyle A}$. The procedure:

1. Toss coins ${\displaystyle A,B,C}$ independently.
2. Apply C-NOT on ${\displaystyle B,C}$.
3. Use coin ${\displaystyle B}$ for conditioning C-SWAP of coins ${\displaystyle A,C}$.

After this procedure, the average (expected value) of the bias of coin ${\displaystyle A}$ is, to leading order, ${\textstyle \varepsilon _{\text{new}}^{\text{average}}={\frac {3}{2}}\varepsilon }$.^[14]

Coins ${\displaystyle B,C}$ are used for C-NOT operation, also known as XOR (exclusive or). The operation is applied in the following manner: ${\displaystyle A_{\text{new}}=A,B_{\text{new}}=B\oplus C,C_{\text{new}}=C}$, which means that ${\displaystyle B\oplus C}$ is computed and replaces the old value of ${\displaystyle B}$, and ${\displaystyle A,C}$ remain unchanged. More specifically, the following operation is applied:

• If the result of coin ${\displaystyle C}$ is 1:
  • Flip coin ${\displaystyle B}$ without looking at the result
• Else (the result of coin ${\displaystyle C}$ is 0):
  • Do nothing (still without looking at the result of ${\displaystyle B}$)

Now, the result of coin ${\displaystyle B_{\text{new}}}$ is checked (without looking at ${\displaystyle A_{\text{new}},C_{\text{new}}}$). Classically, this means that the result of coin ${\displaystyle C}$ must be "forgotten" (cannot be used anymore). This is somewhat problematic classically, because the result of coin ${\displaystyle C}$ is no longer probabilistic; however, the equivalent quantum operators (which are the ones that are actually used in realizations and implementations of the algorithm) are capable of doing so.^[14]

After the C-NOT operation is over, the bias of coin ${\displaystyle C_{\text{new}}}$ is computed using conditional probability:

1. If ${\displaystyle B_{\text{new}}=0}$ (meaning ${\displaystyle B=C}$): $${\displaystyle P(C_{\text{new}}=0|B=C)={\frac {P(B=C=0)}{P(B=C)}}={\frac {\frac {(1+\varepsilon )^{2}}{4}}{{\frac {(1+\varepsilon )^{2}}{4}}+{\frac {(1-\varepsilon )^{2}}{4}}}}={\frac {\frac {(1+\varepsilon )^{2}}{4}}{\frac {1+\varepsilon ^{2}}{2}}}={\frac {1+{\frac {2\varepsilon }{1+\varepsilon ^{2}}}}{2}}.}$$ Therefore, the new bias of coin ${\displaystyle C_{\text{new}}}$ is ${\displaystyle {\frac {2\varepsilon }{1+\varepsilon ^{2}}}}$.
2. If ${\displaystyle B_{\text{new}}=1}$ (meaning ${\displaystyle B\neq C}$): $${\displaystyle P(C_{\text{new}}=0|B\neq C)={\frac {P(C=0,B=1)}{P(B\neq C)}}={\frac {{\frac {1+\varepsilon }{2}}{\frac {1-\varepsilon }{2}}}{{\frac {1+\varepsilon }{2}}{\frac {1-\varepsilon }{2}}+{\frac {1-\varepsilon }{2}}{\frac {1+\varepsilon }{2}}}}={\frac {\frac {1-\varepsilon ^{2}}{4}}{\frac {1-\varepsilon ^{2}}{2}}}={\frac {1}{2}}.}$$ Therefore, the new bias of coin ${\displaystyle C_{\text{new}}}$ is ${\displaystyle 0}$.

Coins ${\displaystyle A_{\text{new}},B_{\text{new}},C_{\text{new}}}$ are used for C-SWAP operation. The operation is applied in the following manner: $${\displaystyle A'=A_{\text{new}}B_{\text{new}}+{\overline {B_{\text{new}}}}C_{\text{new}},\;B'=B_{\text{new}},\;C'=A_{\text{new}}{\overline {B_{\text{new}}}}+B_{\text{new}}C_{\text{new}},}$$ which means that ${\displaystyle A_{\text{new}},C_{\text{new}}}$ are swapped if ${\displaystyle B_{\text{new}}=0}$.

After the C-SWAP operation is over:

1. If ${\displaystyle B_{\text{new}}=0}$: coins ${\displaystyle A_{\text{new}}}$ and ${\displaystyle C_{\text{new}}}$ have been swapped, hence coin ${\displaystyle A'}$ is now ${\textstyle {\frac {2\varepsilon }{1+\varepsilon ^{2}}}}$-biased and coin ${\displaystyle C'}$ is ${\displaystyle \varepsilon }$-biased.
2. Else (${\displaystyle B_{\text{new}}=1}$): coin ${\displaystyle A'}$ remains unchanged (still of bias ${\displaystyle \varepsilon }$) and coin ${\displaystyle C'}$ remains with bias ${\displaystyle 0}$. In this case, coin ${\displaystyle C'}$ can be discarded from the system, as it is too "hot" (its bias is too low, or, equivalently, its entropy is too high).

The average bias of coin ${\displaystyle A'}$ can be calculated by looking at those two cases, using the final bias in each case and the probability of each case: $${\displaystyle \varepsilon _{\text{new}}^{\text{average}}=P(B_{\text{new}}=0)\cdot {\frac {2\varepsilon }{1+\varepsilon ^{2}}}+P(B_{\text{new}}=1)\cdot \varepsilon =\left({\frac {(1+\varepsilon )^{2}}{4}}+{\frac {(1-\varepsilon )^{2}}{4}}\right)\cdot {\frac {2\varepsilon }{1+\varepsilon ^{2}}}+\left({\frac {1+\varepsilon }{2}}{\frac {1-\varepsilon }{2}}+{\frac {1-\varepsilon }{2}}{\frac {1+\varepsilon }{2}}\right)\cdot \varepsilon ={\frac {3\varepsilon }{2}}-{\frac {\varepsilon ^{3}}{2}}}$$ Using the approximation ${\displaystyle \varepsilon \ll 1}$, the new average bias of coin ${\displaystyle A'}$ is ${\textstyle \varepsilon _{\text{new}}^{\text{average}}={\frac {3}{2}}\varepsilon }$. Therefore, these two steps increase the polarization of coin ${\displaystyle A}$ on average.

The algorithm can be written using quantum operations on qubits, as opposed to the classical treatment. In particular, the C-NOT and C-SWAP steps can be replaced by a single unitary quantum operator that operates on the 3 qubits.^[14] Although this operation changes qubits ${\displaystyle B,C}$ in a different manner than the two classical steps, it yields the same final bias for qubit ${\displaystyle A}$. The operator ${\displaystyle U}$ can be uniquely defined by its action on the computational basis of the Hilbert space of 3 qubits:

• ${\displaystyle |000\rangle \mapsto |000\rangle ,}$
• ${\displaystyle |001\rangle \mapsto |001\rangle ,}$
• ${\displaystyle |010\rangle \mapsto |010\rangle ,}$
• ${\displaystyle |011\rangle \mapsto |100\rangle ,}$
• ${\displaystyle |100\rangle \mapsto |011\rangle ,}$
• ${\displaystyle |101\rangle \mapsto |101\rangle ,}$
• ${\displaystyle |110\rangle \mapsto |110\rangle ,}$
• ${\displaystyle |111\rangle \mapsto |111\rangle .}$

In matrix form, this operator is the identity matrix of size 8, except that the 4th and 5th rows are swapped. The result of this operation can be obtained by writing the product state of the 3 qubits, ${\displaystyle \rho _{A,B,C}=\rho _{A}\otimes \rho _{B}\otimes \rho _{C}}$, and applying ${\displaystyle U}$ on it. Afterwards, the bias of qubit ${\displaystyle A}$ can be calculated by projecting its state on the state ${\displaystyle |0\rangle }$ (without projecting qubits ${\displaystyle B,C}$) and taking the trace of the result (see density matrix measurement): $${\displaystyle {\frac {1+\varepsilon _{\text{new}}^{\text{average}}}{2}}=\operatorname {tr} [(P_{0}\otimes I\otimes I)(U\rho _{A,B,C}U^{\dagger })]={\frac {1+{\frac {3\varepsilon }{2}}-{\frac {\varepsilon ^{3}}{2}}}{2}},}$$ where ${\displaystyle P_{0}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}=|0\rangle \langle 0|}$ is the projection on the state ${\displaystyle |0\rangle }$.

Again, using the approximation ${\displaystyle \varepsilon \ll 1}$, the new average bias of coin ${\displaystyle A}$ is ${\textstyle \varepsilon _{\text{new}}^{\text{average}}={\frac {3}{2}}\varepsilon }$.

The irreversible case is an extension of the reversible case: it uses the reversible algorithm as a subroutine. The irreversible algorithm contains another procedure called "Refresh"^[4][14] and extends the reversible one by using a heat bath. This allows for cooling certain qubits (called "reset qubits") without affecting the others, which results in an overall cooling of all the qubits as a system. The cooled reset qubits are used for cooling the rest (called "computational qubits") by applying a compression on them which is similar to the basic compression subroutine from the reversible case. The "insulation" of the computational qubits from the heat bath is a theoretical idealization that does not always hold when implementing the algorithm. However, with a proper choice of the physical implementation of each type of qubit, this assumption fairly holds.^[1][15]

There are many different versions of this algorithm, with different uses of the reset qubits and different achievable biases.^{[1][2][14][7][15]} The common idea behind them can be demonstrated using three qubits: two computational qubits ${\displaystyle A,B}$ and one reset qubit ${\displaystyle C}$.^[4]

Each of the three qubits is initially in a completely mixed state with bias ${\displaystyle 0}$ (see the background section). The following steps are then applied:

1. Refresh: the reset qubit ${\displaystyle C}$ interacts with the heat bath.
2. Compression: a reversible compression (entropy transfer) is applied on the three qubits.

Each round of the algorithm consists of three iterations, and each iteration consists of these two steps (refresh, and then compression). The compression step in each iteration is slightly different, but its goal is to sort the qubits in descending order of bias, so that the reset qubit would have the smallest bias (namely, the highest temperature) of all qubits. This serves two goals:

• Transferring as much entropy as possible away from the computational qubits.
• Transferring as much entropy as possible away from the whole system (and in particular the reset qubit) and into the bath in the following refresh step.

When writing the density matrices after each iteration, the compression step in the 1st round can be effectively treated as follows:

• 1st iteration: swap qubit ${\displaystyle A}$ with the previously refreshed reset qubit ${\displaystyle C}$.
• 2nd iteration: swap qubit ${\displaystyle B}$ with the previously refreshed reset qubit ${\displaystyle C}$.
• 3rd iteration: boost the bias of qubit ${\displaystyle A}$.

The description of the compression step in the following rounds depends on the state of the system before the round has begun and may be more complicated than the above description. In this illustrative description of the algorithm, the boosted bias of qubit ${\displaystyle A}$ (obtained after the end of the first round) is ${\textstyle {\frac {3\varepsilon _{b}}{2}}-{\frac {\varepsilon _{b}^{3}}{2}}}$, where ${\displaystyle \varepsilon _{b}}$ is the bias of the qubits within the heat bath.^[4] This result is obtained after the last compression step; just before this step, the qubits were each ${\displaystyle \varepsilon _{b}}$-biased, which is exactly the state of the qubits before the reversible algorithm is applied.

The contact that is established between the reset qubit and the heat bath can be modeled in several possible ways:

1. A physical interaction between two thermodynamic systems, which eventually results in a reset qubit whose temperature is identical to the bath temperature (equivalently - with bias equal to the bias of the qubits in the bath, ${\displaystyle \varepsilon _{b}}$).
2. A mathematical trace-out on the reset qubit, followed by taking the system in a product state with a fresh new qubit from the bath. This means that we lose the former reset qubit and gain a refreshed new one. Formally, this can be written as ${\displaystyle \rho _{\text{new}}=\operatorname {tr} _{C}(\rho )\otimes \rho _{\varepsilon _{b}}}$, where ${\displaystyle \rho _{\text{new}}}$ is the new density matrix (after the operation is held), ${\displaystyle \operatorname {tr} _{C}(\rho )}$ is the partial trace operation on the reset qubit ${\displaystyle C}$, and ${\displaystyle \rho _{\varepsilon _{b}}}$ is the density matrix describing a (new) qubit from the bath, with bias ${\displaystyle \varepsilon _{b}}$.

In both ways, the result is a reset qubit whose bias is identical to the bias of the qubits in the bath. In addition, the resulted reset qubit is uncorrelated with the other ones, independently of the correlations between them before the refresh step was held. Therefore, the refresh step can be viewed as discarding the information about the current reset qubit and gaining information about a fresh new one from the bath.

The goal of this step is to reversibly redistribute the entropy of all qubits, such that the biases of the qubits are in descending (or non-ascending) order. The operation is done reversibly in order to prevent the entropy of the entire system from increasing (as it cannot decrease in a closed system, see entropy). In terms of temperature, this step rearranges the qubits in ascending order of temperature, so that the reset qubits are the hottest. In the example of the three qubits ${\displaystyle A,B,C}$, this means that after the compression is done, the bias of qubit ${\displaystyle A}$ is the highest and the bias of ${\displaystyle C}$ is the lowest. In addition, the compression is used for the cooling of the computational qubits.

The state of the system will be denoted by ${\displaystyle (\varepsilon _{A},\varepsilon _{B},\varepsilon _{C})}$ if the qubits ${\displaystyle A,B,C}$ are uncorrelated with each other (namely, if the system is in a product state) and their corresponding biases are ${\displaystyle \varepsilon _{A},\varepsilon _{B},\varepsilon _{C}}$.

The compression can be described as a sort operation on the diagonal entries of the density matrix which describes the system. For instance, if the state of the system after a certain reset step is ${\displaystyle (2\varepsilon _{b},0,\varepsilon _{b})}$, then the compression operates on the state as follows:

$${\displaystyle \rho _{ABC}={\frac {1}{8}}\operatorname {diag} {\begin{pmatrix}(1+2\varepsilon _{b})(1+\varepsilon _{b})\\(1+2\varepsilon _{b})(1-\varepsilon _{b})\\(1+2\varepsilon _{b})(1+\varepsilon _{b})\\(1+2\varepsilon _{b})(1-\varepsilon _{b})\\(1-2\varepsilon _{b})(1+\varepsilon _{b})\\(1-2\varepsilon _{b})(1-\varepsilon _{b})\\(1-2\varepsilon _{b})(1+\varepsilon _{b})\\(1-2\varepsilon _{b})(1-\varepsilon _{b})\end{pmatrix}}\xrightarrow {\text{compression}} \rho _{ABC}'={\frac {1}{8}}\operatorname {diag} {\begin{pmatrix}(1+2\varepsilon _{b})(1+\varepsilon _{b})\\(1+2\varepsilon _{b})(1+\varepsilon _{b})\\(1+2\varepsilon _{b})(1-\varepsilon _{b})\\(1+2\varepsilon _{b})(1-\varepsilon _{b})\\(1-2\varepsilon _{b})(1+\varepsilon _{b})\\(1-2\varepsilon _{b})(1+\varepsilon _{b})\\(1-2\varepsilon _{b})(1-\varepsilon _{b})\\(1-2\varepsilon _{b})(1-\varepsilon _{b})\end{pmatrix}}}$$

This notation denotes a diagonal matrix whose diagonal entries are listed within the parentheses. The density matrices ${\displaystyle \rho _{ABC},\rho _{ABC}'}$ represent the state of the system (including possible correlations between the qubits) before and after the compression step, respectively. In the above notations, the state after compression is ${\displaystyle (2\varepsilon _{b},\varepsilon _{b},0)}$.

This sort operation is used for the rearrangement of the qubits in descending order of bias.^[15][4] As in the example, for some cases the sort operation can be described by a simpler operation, such as swap. However, the general form of the compression operation is a sort operation on the diagonal entries of the density matrix.

For an intuitive demonstration of the compression step, the flow of the algorithm in the 1st round is presented below:

• 1st Iteration:
  • After the refresh step, the state is ${\displaystyle (0,0,\varepsilon _{b})}$.
  • After the compression step (which swaps qubits ${\displaystyle A,C}$), the state is ${\displaystyle (\varepsilon _{b},0,0)}$.
• 2nd Iteration:
  • After the refresh step, the state is ${\displaystyle (\varepsilon _{b},0,\varepsilon _{b})}$.
  • After the compression step (which swaps qubits ${\displaystyle B,C}$), the state is ${\displaystyle (\varepsilon _{b},\varepsilon _{b},0)}$.
• 3rd Iteration:
  • After the refresh step, the state is ${\displaystyle (\varepsilon _{b},\varepsilon _{b},\varepsilon _{b})}$.
  • After the compression step (which boosts the bias of qubit ${\displaystyle A}$), the biases of the qubits are ${\textstyle {\frac {3\varepsilon _{b}}{2}}-{\frac {\varepsilon _{b}^{3}}{2}},{\frac {\varepsilon _{b}}{2}}+{\frac {\varepsilon _{b}^{3}}{2}},{\frac {\varepsilon _{b}}{2}}+{\frac {\varepsilon _{b}^{3}}{2}}}$, which can be approximated (to leading order) by ${\textstyle {\frac {3\varepsilon _{b}}{2}},{\frac {\varepsilon _{b}}{2}},{\frac {\varepsilon _{b}}{2}}}$. Here, each bias is independently defined as the bias of the matching qubit when discarding the rest of the system (using partial trace), even when there are correlations between them. Therefore, this notation cannot fully describe the system, but can only be used as an intuitive demonstration of the steps of the algorithm.

After the 1st round is over, the bias of the reset qubit (${\textstyle {\frac {\varepsilon _{b}}{2}}+{\frac {\varepsilon _{b}^{3}}{2}}}$) is smaller than the bias of the heat bath (${\displaystyle \varepsilon _{b}}$). This means that in the next refresh step (in the 2nd round of the algorithm), the reset qubit will be replaced by a fresh qubit with bias ${\displaystyle \varepsilon _{b}}$: this cools the entire system, similarly to the previous refresh steps. Afterwards, the algorithm continues in a similar way.

The number of rounds is not bounded: since the biases of the reset qubits asymptotically reach the bias of the bath after each round, the bias of the target computational qubit asymptotically reaches its limit as the algorithm proceeds.^[2][15] The target qubit is the computational qubit that the algorithm aims to cool the most. The "cooling limit" (the maximum bias the target qubit can reach) depends on the bias of the bath and the number of qubits of each kind in the system. If the number of the computational qubits (excluding the target one) is ${\displaystyle n'}$ and the number of reset qubits is ${\displaystyle m}$, then the cooling limit is ${\textstyle \varepsilon _{\max }={\frac {(1+\varepsilon _{b})^{m2^{n'}}-(1-\varepsilon _{b})^{m2^{n'}}}{(1+\varepsilon _{b})^{m2^{n'}}+(1-\varepsilon _{b})^{m2^{n'}}}}}$.^[4] In the case where ${\displaystyle \varepsilon _{b}\ll m2^{n'}}$, the maximal polarization that can be obtained is proportional to ${\displaystyle m2^{n'}}$. Otherwise, the maximal bias reaches arbitrarily close to ${\displaystyle 1}$. The number of rounds required in order to reach a certain bias depends on the desired bias, the bias of the bath and the number of qubits, and moreover varies between different versions of the algorithm.^[16][4][1]

There are other theoretical results which give bounds on the number of iterations required to reach a certain bias. For example, if the bias of the bath is ${\displaystyle \varepsilon _{b}\ll 1}$, then the number of iterations required to cool a certain qubit to bias ${\displaystyle k\varepsilon _{b}\ll 1}$ is at least ${\displaystyle k^{2}}$.