Thermodynamics

Algorithmic cooling can be discussed using classical and quantum thermodynamics points of view.

Quantum mechanics

Polarization or bias of a state

The state ${\displaystyle \rho }$ above is called ${\displaystyle \varepsilon }$-polarized, or ${\displaystyle \varepsilon }$-biased, since it deviates by ${\displaystyle \varepsilon }$ in the diagonal entries from the completely mixed state.

Another approach for the definition of bias or polarization is using Bloch sphere (or generally Bloch ball). Restricted to a diagonal density matrix, a state can be on the straight line connecting the antipodal points representing the states ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ ("north and south poles" of the sphere). In this approach, the ${\displaystyle \varepsilon }$ parameter (${\displaystyle \varepsilon \in [-1,1]}$) is exactly the distance (up to a sign) of the state from the center of the ball, which represents the completely mixed state. For ${\displaystyle \varepsilon =\pm 1}$ the state is exactly on the poles and for ${\displaystyle \varepsilon =0}$ the state is exactly in the center. A bias can be negative (for example ${\textstyle -{\frac {1}{2}}}$), and in this case the state is in the middle between the center and the south pole.

In the Pauli matrices representation form, an ${\displaystyle \varepsilon }$-biased quantum state is^[4]

$${\displaystyle \rho ={\frac {1}{2}}{\begin{pmatrix}1+\varepsilon &0\\0&1-\varepsilon \end{pmatrix}}={\frac {1}{2}}\left(I+(0,0,\varepsilon )\cdot {\vec {\sigma }}\right)={\frac {1}{2}}(I+\varepsilon \sigma _{z}).}$$

Entropy

Since quantum systems are involved, the entropy used here is von Neumann entropy. For a single qubit represented by the (diagonal) density matrix above, its entropy is

$${\displaystyle H(\varepsilon )=-\left({\frac {1+\varepsilon }{2}}\log {\frac {1+\varepsilon }{2}}+{\frac {1-\varepsilon }{2}}\log {\frac {1-\varepsilon }{2}}\right)}$$

(where the logarithm is to base ${\displaystyle 2}$). This expression coincides with the entropy of an unfair coin with "bias" ${\displaystyle \varepsilon }$, meaning probability ${\textstyle {\frac {1+\varepsilon }{2}}}$ for tossing heads. A coin with bias ${\displaystyle \varepsilon =\pm 1}$ is deterministic with zero entropy, and a coin with bias ${\displaystyle \varepsilon =0}$ is fair with maximal entropy (${\displaystyle H(\varepsilon =0)=\log 2=1)}$.

The relation between the coins approach and von Neumann entropy is an example of the relation between entropy in thermodynamics and in information theory.

Physics

The physical intuition for this family of algorithms comes from classical thermodynamics.^[3]

Information theory

The intuition for this family of algorithms can come from an extension of Von-Neumann's solution for the problem of obtaining fair results from a biased coin.^[5] In this approach to algorithmic cooling, the bias of the qubits is merely a probability bias, or the "unfairness" of a coin.

Quantum error correction

Quantum error correction is a quantum algorithm for protection from errors. The algorithm operates on the relevant qubits (which operate within the computation) and needs a supply of new pure qubits for each round. This requirement can be weakened^[4][8] to purity above a certain threshold instead of requiring fully pure qubits. For this, algorithmic cooling can be used to produce qubits with the desired purity for quantum error correction.

Ensemble computing

Ensemble computing is a computational model that uses a macroscopic number of identical computers. Each computer contains a certain number of qubits, and the computational operations are performed simultaneously on all the computers. The output of the computation can be obtained by measuring the state of the entire ensemble, which would be the average output of each computer in it.^[9] Since the number of computers is macroscopic, the output signal is easier to detect and measure than the output signal of each single computer.

This model is widely used in NMR quantum computing: each computer is represented by a single (identical) molecule, and the qubits of each computer are the nuclear spins of its atoms. The obtained (averaged) output is a detectable magnetic signal.

NMR spectroscopy

Nuclear magnetic resonance spectroscopy (sometimes called MRS - magnetic resonance spectroscopy) is a non-invasive technique related to MRI (magnetic resonance imaging) for analyzing metabolic changes in vivo (from Latin: "within the living organism"), which can potentially be used for diagnosing brain tumors, Parkinson's disease, depression, etc. It uses some magnetic properties of the relevant metabolites to measure their concentrations in the body, which are correlated with certain diseases. For example, the difference between the concentrations of the metabolites glutamate and glutamine can be linked to some stages of neurodegenerative diseases, such as Alzheimer's disease.^[10]

Some uses of MRS focus on the carbon atoms of the metabolites (see carbon-13 nuclear magnetic resonance). One major reason for this is the presence of carbon in a large portion of all tested metabolites. Another reason is the ability to mark certain metabolites by the ¹³C isotope, which is more easy to measure than the usually used hydrogen atoms mainly because of its magnetic properties (such as its gyromagnetic ratio).

In MRS, the nuclear spins of the atoms of the metabolites are required to be with a certain degree of polarization, so the spectroscopy can succeed. Algorithmic cooling can be applied^[7] in vivo, increasing the resolution and precision of the MRS. Realizations (not in vivo) of algorithmic cooling on metabolites with ¹³C isotope have been shown to increase the polarization of ¹³C in amino acids^[11] and other metabolites.^[12][13]

MRS can be used to obtain biochemical information about certain body tissues in a non-invasive manner. This means that the operation must be carried out at room temperature. Some methods of increasing polarization of spins (such as hyperpolarization, and in particular dynamic nuclear polarization) are not able to operate under this condition since they require a cold environment (a typical value is 1K, about -272 degrees Celsius). On the other hand, algorithmic cooling can be operated in room temperature and be used in MRS in vivo,^[7] while methods that required lower temperature can be used in biopsy, outside of the living body.

C-NOT step

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 ${\displaystyle 0}$.

C-SWAP step

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.

Alternative explanation: quantum operations

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 ${\textstyle \varepsilon _{\text{new}}^{\text{average}}={\frac {3}{2}}\varepsilon }$.

Refresh step

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.

Compression step

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.

General results

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 ${\displaystyle k^{2}}$.